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@#HHHۃHH=+t~HHH͢HH谢HH蓢HHۃHL$ILLULH#HlLH#EHtYL1H?H	PH?H!UHBHH+H@L#(IHIH$LH#H9}H<H&'HW(L0zI<H߀H='+HEHHǀHHE軀H=+HEH7HE HH$dH3%(tH[]A\A]A^A_HH0HH%s not implemented on this architecturePARI: %.*s%*s%.*s%*s%sPARI: %.*s%*s%sPARI: %sAttempt to ask Perl to free PARI function not installed from Perlarg1, arg2XSUB call through interface did not provide *functionarg1, arg2, arg3arg1a, b, c=0Corrupted data: should be variablepanic: PARI narg value not attachedMath::PariMath::Pari::EpBad PARI variable name "%s" specifiedintiter%i<null>name, v = 99gcmp1gcmp_1gdiventgdivroundgegalgequalgpow_gadd_gand_gbitand_gbitor_gbitxor_gbitneg_gbitshiftl_gcmp_gcmp0_gdiv_geq_gge_ggt_gle_glt_gmul_gmod_gneg_gne_gor_gpui_gsub_abs_cos_lex_log_sin`%s' is not a Pari function namevGVlGGvLGvLLGDVDEGDVDIV=GEpV=GIpvV=GGvLGGGV=GGEGDGDGDGGD0,LLGD0,LvV=GGGV=GGEDV=GGIDGGDVDVDGD0,L,DGD0,L,DGpGGGD0,L,pvV=GED0,L,vV=GID0,L,GDGDGD0,L,pGD0,L,D0,G,GD0,G,D0,G,D0,L,pLV=GGEpD0,L,D0,L,LV=GGIpD0,L,D0,L,Do not know how to work with Pari control structure `%s'Unsupported Pari function %s, interface 0 code NULLUnsupported interface %ld for "direct-link" Pari function %sUnsupported interface %ld and no code for a Pari function %sinterfacePari.xsCannot load a Pari macro `%s': macros are unsupported; VALENCE=%#04x, code=<%s>, isFunction=%d, EpVAR=%dheap had %ld bytes (%ld items)
arg1, arg2, arg3, arg4Got the type 0x%x instead of CV=0x%x or GV=0x%x in %s, %iSomething very wrong  in %s, %iThe longword %ld ordinal out of boundInternal error in sv2pari!Variable in sv2pari is not of known typex, nname, valarg1, arg2, invarg1, arg2, arg3, arg4=0arg1, arg2, arg3, arg4, arg5arg1, arg2, arg3, arg4, arg5, arg6=0, arg7=0arg1, arg2, arg3=0arg0, arg00, arg1=0, arg2=0, arg3=0Same iterator for a double looparg1, arg2, arg3, arg4, arg0=0arg1, arg2=0, arg3=0, arg4=0arg1, arg2=0, arg3=0arg1, arg2=0%ld'th argument (of %ld) to PARImatL() is not a vectorin, dummy1, dummy2svg, eltg, n, eltAccess to elements of not-a-vectorArray index %li out of rangeNot a vector where column of a matrix expectedAssignment of a columns into a matrix of incompatible heightg, nXSUB call through interface with a NULL codeToo many args for a flexible-interface functionDid not get a variablepanic: no arg when AssignPariExpr()More than 2 running variables per PARI entry point not supportedType E, I without a preceding variableUnexpected syntax of default argument '%s' in prototype '%s'Cannot process default argument %.*s of type %.1s for prototype '%s'Calling Perl via PARI with an unknown interface: avoiding loopUnsupported code '%.1s' in signature '%s' of a PARI function `%s'Too few args %d for PARI function `%s'%d unused args for PARI function %s of signature `%s' (with %d args)Columns of input matrix are of different heightNot a matrix where matrix expected%ld'th argument (of %ld) to PARImat() is not a vectorin, ...digits=0n = 0bernzone# %sstack size is %ld bytes (%ld x %ld longs)
%s %2ld: %s
%sour data takes %ld words out of %ld words on stack
Expected int return type, got code '%s'Expected long return type, got code '%s'Expected GEN return type, got code '%s'Expected VOID return type, got code '%s'%li items moved off stack, onStack=%ld, offStack=%ldPerl function exported into PARI returns unexpected number %d of values (need %d)Can't install Perl function with prototype `%s'Import of Perl function with too many argumentsMath::Pari::convertedcv, name, numargs = 1, help = NULLnewsize = 0newvalue = -1tag2.030523v5.36.0Pari.cMath::Pari::FETCHMath::Pari::STOREMath::Pari::FETCHSIZEMath::Pari::EXISTSMath::Pari::is_gnilMath::Pari::sv2pariMath::Pari::sv2parimatMath::Pari::pari2ivMath::Pari::pari2nv$;@Math::Pari::pari2num_Math::Pari::pari2numMath::Pari::pari2pvMath::Pari::_to_intMath::Pari::PARIMath::Pari::PARIcolMath::Pari::PARIvecLMath::Pari::PARIcolLMath::Pari::PARImatMath::Pari::PARImatLMath::Pari::installPerlFunctionCVMath::Pari::interface_flexible_voidMath::Pari::interface_flexible_genMath::Pari::interface_flexible_longMath::Pari::interface_flexible_intMath::Pari::interface0Math::Pari::interface9900Math::Pari::interface1Math::Pari::interface199Math::Pari::interface10Math::Pari::interface109Math::Pari::interface11Math::Pari::interface15Math::Pari::interface18Math::Pari::interface2Math::Pari::interface299Math::Pari::interface20Math::Pari::interface2099Math::Pari::interface209Math::Pari::interface2091Math::Pari::interface29Math::Pari::interface3Math::Pari::interface30Math::Pari::interface4Math::Pari::interface5Math::Pari::interface12Math::Pari::interface13Math::Pari::interface14Math::Pari::interface21Math::Pari::interface2199Math::Pari::interface22Math::Pari::interface23Math::Pari::interface24Math::Pari::interface25Math::Pari::interface26Math::Pari::interface27Math::Pari::interface28Math::Pari::interface28_oldMath::Pari::interface29_oldMath::Pari::interface31Math::Pari::interface32Math::Pari::interface33Math::Pari::interface34Math::Pari::interface35Math::Pari::interface37$$$$;$Math::Pari::interface47Math::Pari::interface48$$;$$$Math::Pari::interface49Math::Pari::interface83Math::Pari::interface84Math::Pari::interface16Math::Pari::interface19Math::Pari::interface44Math::Pari::interface45$$$$$Math::Pari::interface59$$$$$;$$Math::Pari::interface73Math::Pari::interface86Math::Pari::interface87Math::Pari::_2boolMath::Pari::pari2boolMath::Pari::loadPariMath::Pari::_listPariMath::Pari::memUsageMath::Pari::dumpStackMath::Pari::__dumpStackMath::Pari::dumpHeapMath::Pari::DESTROYMath::Pari::pari_printMath::Pari::pari_pprintMath::Pari::pari_texprintMath::Pari::typMath::Pari::PARIvarMath::Pari::ifactMath::Pari::changevalueMath::Pari::set_gnutermMath::Pari::setprecisionMath::Pari::setseriesprecisionMath::Pari::setprimelimitMath::Pari::int_set_term_ftableMath::Pari::pari_version_expMath::Pari::have_highlevelMath::Pari::have_graphicsMath::Pari::PARI_DEBUGMath::Pari::PARI_DEBUG_setMath::Pari::lgefMath::Pari::lgefintMath::Pari::lgMath::Pari::longwordMath::Pari::added_sectionsMath::Pari::__detach_stackMath::Pari::type_nameMath::Pari::reset_on_reloadMath::Pari::allocatememMath::Pari::initmemMath::Pari::initprimes$Math::Pari::initmem not defined!$Math::Pari::initprimes not defined!gRv hhhRv#iRvRvRvRv]iiij.jRvRvajljj+kkkkvkvvvvk%l_lllvvlbvvwxxTyyz{r{{^|^|^|^|^||xD0,G,D0,G,D0,G,D0,G,D0,G,D0,G,%s

couldn't open dynamic library '%s'couldn't open dynamic symbol table of processcan't find symbol '%s' in library '%s'can't find symbol '%s' in dynamic symbol table of process[secure mode]: about to install '%s'. OK ? (^C if not)
installvrrD"",r,D"",s,install(name,code,{gpname},{lib}): load from dynamic library 'lib' the function 'name'. Assign to it the name 'gpname' in this GP session, with argument code 'code'. If 'lib' is omitted use 'libpari.so'. If 'gpname' is omitted, use 'name'vV=GGIDGDGpplot(X=a,b,expr,{ymin},{ymax}): crude plot of expression expr, X goes from a to b, with Y ranging from ymin to ymax. If ymin (resp. ymax) is not given, the minima (resp. the maxima) of the expression is used insteadplotboxvLGGplotbox(w,x2,y2): if the cursor is at position (x1,y1), draw a box with diagonal (x1,y1) and (x2,y2) in rectwindow w (cursor does not move)plotclipvLplotclip(w): clip the contents of the rectwindow to the bounding box (except strings)plotcolorplotcolor(w,c): in rectwindow w, set default color to c. Possible values for c are 1=black, 2=blue, 3=sienna, 4=red, 5=cornsilk, 6=grey, 7=gainsboroughplotcopyvLLGGD0,L,plotcopy(sourcew,destw,dx,dy,{flag=0}): copy the contents of rectwindow sourcew to rectwindow destw with offset (dx,dy). If flag's bit 1 is set, dx and dy express fractions of the size of the current output device, otherwise dx and dy are in pixels. dx and dy are relative positions of northwest corners if other bits of flag vanish, otherwise of: 2: southwest, 4: southeast, 6: northeast cornersplotcursorplotcursor(w): current position of cursor in rectwindow wplotdrawvGD0,L,plotdraw(list, {flag=0}): draw vector of rectwindows list at indicated x,y positions; list is a vector w1,x1,y1,w2,x2,y2,etc. . If flag!=0, x1, y1 etc. express fractions of the size of the current output deviceV=GGIpD0,M,D0,L,
Parametric|1; Recursive|2; no_Rescale|4; no_X_axis|8; no_Y_axis|16; no_Frame|32; no_Lines|64; Points_too|128; Splines|256; no_X_ticks|512; no_Y_ticks|1024; Same_ticks|2048ploth(X=a,b,expr,{flags=0},{n=0}): plot of expression expr, X goes from a to b in high resolution. Both flags and n are optional. Binary digits of flags mean: 1=Parametric, 2=Recursive, 4=no_Rescale, 8=no_X_axis, 16=no_Y_axis, 32=no_Frame, 64=no_Lines (do not join points), 128=Points_too (plot both lines and points), 256=Splines (use cubic splines), 512=no_X_ticks, 1024= no_Y_ticks, 2048=Same_ticks (plot all ticks with the same length). n specifies number of reference points on the graph (0=use default value). Returns a vector for the bounding boxplothraw(listx,listy,{flag=0}): plot in high resolution points whose x (resp. y) coordinates are in listx (resp. listy). If flag is 1, join points, other non-0 flags should be combinations of bits 8,16,32,64,128,256 meaning the same as for ploth()plothsizesplothsizes({flag=0}): returns array of 6 elements: terminal width and height, sizes for ticks in horizontal and vertical directions, width and height of characters. If flag=0, sizes of ticks and characters are in pixels, otherwise are fractions of the screen sizeplotinitvLD0,G,D0,G,D0,L,plotinit(w,{x=0},{y=0},{flag=0}): initialize rectwindow w to size x,y. If flag!=0, x and y express fractions of the size of the current output device. x=0 or y=0 means use the full size of the deviceplotkillplotkill(w): erase the rectwindow wplotlinesvLGGD0,L,plotlines(w,listx,listy,{flag=0}): draws an open polygon in rectwindow w where listx and listy contain the x (resp. y) coordinates of the vertices. If listx and listy are both single values (i.e not vectors), draw the corresponding line (and move cursor). If (optional) flag is non-zero, close the polygonplotlinetypeplotlinetype(w,type): change the type of following lines in rectwindow w. type -2 corresponds to frames, -1 to axes, larger values may correspond to something else. w=-1 changes highlevel plottingplotmoveplotmove(w,x,y): move cursor to position x,y in rectwindow wplotpointsplotpoints(w,listx,listy): draws in rectwindow w the points whose x (resp y) coordinates are in listx (resp listy). If listx and listy are both single values (i.e not vectors), draw the corresponding point (and move cursor)plotpointsizeplotpointsize(w,size): change the "size" of following points in rectwindow w. w=-1 changes global valueplotpointtypeplotpointtype(w,type): change the type of following points in rectwindow w. type -1 corresponds to a dot, larger values may correspond to something else. w=-1 changes highlevel plottingplotrboxplotrbox(w,dx,dy): if the cursor is at (x1,y1), draw a box with diagonal (x1,y1)-(x1+dx,y1+dy) in rectwindow w (cursor does not move)plotrecthplotrecth(w,X=xmin,xmax,expr,{flags=0},{n=0}): plot graph(s) for expr in rectwindow w, where expr is scalar for a single non-parametric plot, and a vector otherwise. If plotting is parametric, its length should be even and pairs of entries give points coordinates. If not, all entries but the first are y-coordinates. Both flags and n are optional. Binary digits of flags mean: 1 parametric plot, 2 recursive plot, 4 do not rescale w, 8 omit x-axis, 16 omit y-axis, 32 omit frame, 64 do not join points, 128 plot both lines and points. n specifies the number of reference points on the graph (0=use default value). Returns a vector for the bounding boxplotrecthrawLGD0,L,plotrecthraw(w,data,{flags=0}): plot graph(s) for data in rectwindow w, where data is a vector of vectors. If plot is parametric, length of data should be even, and pairs of entries give curves to plot. If not, first entry gives x-coordinate, and the other ones y-coordinates. Admits the same optional flags as plotrecth, save that recursive plot is meaninglessplotrlineplotrline(w,dx,dy): if the cursor is at (x1,y1), draw a line from (x1,y1) to (x1+dx,y1+dy) (and move the cursor) in the rectwindow wplotrmoveplotrmove(w,dx,dy): move cursor to position (dx,dy) relative to the present position in the rectwindow wplotrpointplotrpoint(w,dx,dy): draw a point (and move cursor) at position dx,dy relative to present position of the cursor in rectwindow wplotscalevLGGGGplotscale(w,x1,x2,y1,y2): scale the coordinates in rectwindow w so that x goes from x1 to x2 and y from y1 to y2 (y2<y1 is allowed)plotstringvLsD0,L,plotstring(w,x,{flags=0}): draw in rectwindow w the string corresponding to x. Bits 1 and 2 of flag regulate horizontal alignment: left if 0, right if 2, center if 1. Bits 4 and 8 regulate vertical alignment: bottom if 0, top if 8, v-center if 4. Can insert additional gap between point and string: horizontal if bit 16 is set, vertical if bit 32 is setplottermplotterm("termname"): set terminal to plot in high resolution to. Ignored by some drivers. In gnuplot driver possible terminals are the same as in gnuplot, terminal options can be put after the terminal name and space; terminal size can be put immediately after the name, as in "gif=300,200". If term is "?", lists possible values. Positive return value means successpsdrawpsdraw(list, {flag=0}): same as plotdraw, except that the output is a postscript program in psfile (pari.ps by default), and flag!=0 scales the plot from size of the current output device to the standard postscript plotting sizepsplothpsploth(X=a,b,expr,{flags=0},{n=0}): same as ploth, except that the output is a postscript program in psfile (pari.ps by default)psplothrawpsplothraw(listx,listy,{flag=0}): same as plothraw, except that the output is a postscript program in psfile (pari.ps by default)removeprimeprime %Z is not in primetableIFAC: Stop: Primary factor: %Z
IFAC: Stop: remaining %Z
zero argument in an arithmetic functionpanic: set_optimizen-th prime meaningless if n = %ldzero argument in factorintIFAC: (Partial fact.) Initial stop requested.
primepiToo large primelimitaddprimecan't accept 0 in addprimesdenominators not allowed in divisorstoo many divisors (more than %ld)binarynegative exponent in bitwise negationbitwise orbitwise andbitwise xorbitwise negated imply@@q=
ףp?C??p=
ף@?found factor
	%Z
currently lost to the factoring machinerymiller(rabin)[caller of] elladd0SQUFOF: found factor %ld from ambiguous form
	after %ld steps on the ambiguous cycle, time = %ld ms
SQUFOF: ...found nothing on the ambiguous cycle
	after %ld steps there, time = %ld ms
SQUFOF: squfof_ambig returned %ld
factor has NULL exponent in ifac_findRho: time = %6ld ms,	%3ld round%s
Miller-Rabin: testing base %ld
LucasModcompositeavoiding nonexistent factors in ifac_whoiswhoIFAC: factor %Z
	is prime (no larger composite)
IFAC: prime %Z
	appears with exponent = %ld
IFAC: factor %Z
	is %s
PL: proving primality of N = %Z
PL: N-1 factored!
False prime number %Z in plisprimesnextpr: %lu != prc210_rp[%ld] mod 210
[caller of] snextprsnextpr: %lu should have been prime but isn't
snextpr: integer wraparound after prime %lu
ECM: number too small to justify this stage
ECM: working on %ld curves at a time; initializing for one round for up to %ld roundsECM: stack tight, using heap space
ECM: time = %6ld ms
ECM: dsn = %2ld,	B1 = %4lu,	B2 = %6lu,	gss = %4ld*420
ECM: time = %6ld ms, B1 phase done, p = %lu, setting up for B2
	(got [2]Q...[10]Q)
ECM: %lu should have been prime but isn't
ellfacteur	(got [p]Q, p = %lu = prc210_rp[%ld] mod 210)
	(got initial helix)
ECM: time = %6ld ms, entering B2 phase, p = %lu
ECM: finishing curves %ld...%ld
	(extracted precomputed helix / baby step entries)
	(baby step table complete)
	(giant step at p = %lu)
ECM: time = %6ld ms,	ellfacteur giving up.
ECM: time = %6ld ms,	p <= %6lu,
	found factor = %Z
composite Rho: searching small factor of %ld-bit integer
Rho: restarting for remaining rounds...
Rho: using X^2%+1ld for up to %ld rounds of 32 iterations
Rho: time = %6ld ms,	Pollard-Brent giving up.
Rho: fast forward phase (%ld rounds of 64)...
Rho: time = %6ld ms,	%3ld rounds, back to normal mode
Rho: hang on a second, we got something here...
	Pollard-Brent failed.
	found %sfactor = %Z
	found factors = %Z, %Z,
	and %Z
squfof [caller of] (n or 3n is a square)squfof [caller of] (5n is a square)SQUFOF: entering main loop with forms
	(1, %ld, %ld) and (1, %ld, %ld)
	of discriminants
	%Z and %Z, respectively
SQUFOF: blacklisting a = %ld on first cycle
SQUFOF: blacklisting a = %ld on second cycle
SQUFOF: first cycle exhausted after %ld iterations,
	dropping it
SQUFOF: square form (%ld^2, %ld, %ld) on first cycle
	after %ld iterations, time = %ld ms
SQUFOF: found factor %ld^2
SQUFOF: ...but the root form seems to be on the principal cycle
SQUFOF: second cycle exhausted after %ld iterations,
	dropping it
SQUFOF: square form (%ld^2, %ld, %ld) on second cycle
	after %ld iterations, time = %ld ms
SQUFOF: giving up, time = %ld ms
, orOddPwrs: is %Z
	...a 3rd%s 5th%s 7th power?
	modulo: resid. (remaining possibilities)
	   %3ld:  %3ld   (3rd %ld, 5th %ld, 7th %ld)
	But it nevertheless wasn't a %ld%s power.
	checking modulo %ld
	- ruled out
OddPwrs: passed modular checks
	But it wasn't a pure power.
OddPwrs: examining %Z
OddPwrs: testing for exponent %ld
factoring 0 in ifac_startIFAC: new partial factorization structure (%ld slots)
 (so far)...IFAC: main loop: repeated old factor
	%Z
IFAC: cracking composite
	%Z
IFAC: checking for pure square
IFAC: found %Z =
	%Z ^2
IFAC: factor %Z
	is prime
IFAC: checking for odd power
IFAC: found %Z =
	%Z ^%ld
IFAC: trying Pollard-Brent rho method
IFAC: trying Shanks' SQUFOF, will fail silently if input
      is too large for it.
IFAC: trying Lenstra-Montgomery ECM
IFAC: trying MPQS
IFAC: forcing ECM, may take some time
IFAC: unfactored composite declared primeIFAC: untested integer declared primeIFAC: incorporating set of %ld factor(s)
	stored (largest) factor no. %ld...
	factor no. %ld is a duplicate%s
	factor no. %ld was unique%s
IFAC: factoring %Z
	yielded `factor' %Z
	which isn't!
factoringIFAC: cofactor = %Z
ifac_crack [Z_issquarerem miss]IFAC: main loop: repeated new factor
	%Z
IFAC: a factor was a power of another prime factor
IFAC: a factor was divisible by another prime factor,
	leaving a cofactor = %Z
IFAC: prime %Z
	appears at least to the power %ld
IFAC: main loop: another factor was divisible by
	%Z
partial impossibly short in ifac_sort_one`*where' out of bounds in ifac_sort_one`washere' out of bounds in ifac_sort_onemisaligned partial detected in ifac_sort_oneIFAC: repeated factor %Z
	detected in ifac_sort_one
composite equals prime in ifac_sort_oneprime equals composite in ifac_sort_onenon-existent factor class in ifac_mainIFAC: after main loop: repeated old factor
	%Z
IFAC: main loop: %ld factor%s left
IFAC: main loop: this was the last factor
factoring 0 in ifac_decompIFAC: (Partial fact.)Stop requested.
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 !"#$%&'()*+,-.//7?Unsupported function map3d_xy calledExpect a number, got a stringpanic: gnuplotgnuplot-like plotting environment not loaded yetThis runtime link with gnuplot-shim does not implement midlevel start/end functionspanic: more than %d tokens for optionsNo terminal specifiedTerminal does not define optionsCannot reset output routines to copy term list to a variableCannot reset output routines back...Low-level terminals of Gnuplot.  Query available terminals via
	plotterm("??")
Specify size (if it changable via scaling) as in "termname=300,200".
Add terminal options (if applicable) after the name and SPACE char.
See documentation of options in gnuplot, or, if via Term::Gnuplot, via
    perldoc GnuplotTerminals
name "%s" for terminal too longTerminal does not define reseterror setting terminal "%s"Terminal size directive without ','Terminal does not define pointsizeX11dumbDISPLAYTerminal does not define linetypeTerminal does not define pointTerminal does not define moveTerminal does not define vectorTerminal does not define justify_textTerminal does not define put_textGnuplotGNUPLOT_DRAW_DLLGNUPLOT_DRAW_DLL_NO_PERLperl -MConfig -wle %cuse Term::Gnuplot;print $INC{qq(Term/Gnuplot.pm)};print $Config{dlext}%c.pm
filename of Term::Gnuplot does not end in `.pm': `%s'/blib/libarch/auto/Term/Gnuplot/perl -MDynaLoader -we %cpackage DynaLoader; print mod2fname([qw(Term Gnuplot)]) if defined &mod2fname%cBuffer overflow finding gnuplot DLLCan't load Gnuplot drawing engine from '%s': %sget_term_ftableCan't resolve 'get_term_ftable' function from Gnuplot drawing engine '%s': %sCan't find Gnuplot drawing engine DLL,
	set GNUPLOT_DRAW_DLL environment variable to the name of the DLL,
	or install Perl module Term::Gnuplot, e.g., by running
		perl -MCPAN -e "install Term::Gnuplot"
	With Term::Gnuplot, if you don't have root access, consult
		perldoc -q "my own"
	alternatively, you can use an uninstalled version of Term::Gnuplot
	by running GP/PARI as
		env PERL5OPT=-Mblib=/directory/of/build/of/Term-Gnuplot gp
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S@?@@?Uk@MbP?8??0@?&.>$@killing bloc (no %ld): %08lx
popping %s (bloc no %ld)
pari.pspari.logGP_DATA_DIR/usr/local/lib/pari/entering recover(), loc = %ld
leaving recover()
  *** %s: %s  ***   %s in %s; new prec = %ld
 %s: %s
Resetting all trapsbad object %Zbad component %ld in object %Z
For full compatibility with GP 1.39.15, type "default(compatible,3)", or set "compatible = 3" in your GPRC fileadditionmultiplicationdivision-->uncaught error: %ld%s file  ###   user error:  %s is not yet implemented. in %s. %s, please report %s %s %s %s.%lu.
  current stack size: %lu (%.3f Mbytes)
  [hint] you can increase GP stack with allocatemem()
use pari_warn for warningsmallocing NULL objectmallocing NULL object in newblocnew bloc, size %6lu (no %ld): %08lx
  ***   %s: user interruptsegmentation fault: bug in PARI or calling programbus error: bug in PARI or calling programfloating point exception: bug in PARI or calling programbroken pipeunknown signalnot enough memory, new stack %luvariable out of range in reorderduplicate indeterminates in reordercan't trap memory errorsno such error number: %lderrpiletypeergdiverinvmoderaccurerarchersigintertalkeruserthis trap keyword  ***   Error in the PARI system. End of program.
Cannot initialize kernellbot>ltop in gerepilesignificant pointers lost in gerepile! (please report)changevardoubling stack size; new stack = %lu (%.3f Mbytes)bot=0x%lx	top=0x%lx
0x%p:	0x%lx	%lu
Time ColCol({x=[]}): transforms the object x into a column vector. Empty vector if x is omittedEulerEuler=Euler(): Euler's constant with current precisionI=I(): square root of -1List({x=[]}): transforms the vector or list x into a list. Empty list if x is omittedMatMat({x=[]}): transforms any GEN x into a matrix. Empty matrix if x is omittedMod(x,y): creates 'x modulo y'.OO(a^b): p-adic or power series zero with precision given by bPiPi=Pi(): the constant pi, with current precisionPolPol(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientPolrevPolrev(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termGGGDGpQfb(a,b,c,{D=0.}): binary quadratic form a*x^2+b*x*y+c*y^2. D is optional (0.0 by default) and initializes Shanks's distance if b^2-4*a*c>0SerSer(x,{v=x}): convert x (usually a vector) into a power series with variable v, starting with the constant coefficientSetSet({x=[]}): convert x into a set, i.e. a row vector with strictly increasing coefficients. Empty set if x is omittedStrStr({str}*): concatenates its (string) argument into a single stringStrchrStrchr(x): converts x to a string, translating each integer into a characterStrexpandStrexpand({str}*): concatenates its (string) argument into a single string, performing tilde expansionStrtexStrtex({str}*): translates its (string) arguments to TeX format and returns the resulting stringVecVec({x=[]}): transforms the object x into a vector. Empty vector if x is omittedVecsmallVecsmall({x=[]}): transforms the object x into a VECSMALL. Empty vector if x is omittedabs(x): absolute value (or modulus) of xacos(x): inverse cosine of xacoshacosh(x): inverse hyperbolic cosine of xaddhelpvSsaddhelp(symbol,"message"): add/change help message for a symboladdprimes({x=[]}): add primes in the vector x to the prime table to be used in trial division. x may also be a single integer. Composite "primes" are allowed, and in that case you may later get a message "impossible inverse", which will give you some factors. List the current extra primes if x is omitted. If some primes are added which intersect non trivially the existing table entries, suitable updating is doneagm(x,y): arithmetic-geometric mean of x and yGLD0,L,palgdep(x,n,{flag=0}): algebraic relations up to degree n of x, using lindep([1,x,...,x^(n-1)], flag).vrralias("new","old"): new is now an alias for oldvD0,L,allocatemem({s=0}): allocates a new stack of s bytes. doubles the stack if s is omittedarg(x): argument of x,such that -pi<arg(x)<=piasin(x): inverse sine of xasinhasinh(x): inverse hyperbolic sine of xatan(x): inverse tangent of xatanh(x): inverse hyperbolic tangent of xbernfracbernfrac(x): Bernoulli number B_x, as a rational numberbernrealbernreal(x): Bernoulli number B_x, as a real number with the current precisionbernvecbernvec(x): Vector of rational Bernoulli numbers B_0, B_2,...up to B_(2x)besselh1besselh1(nu,x): H^1-bessel function of index nu and argument xbesselh2besselh2(nu,x): H^2-bessel function of index nu and argument xbesselibesseli(nu,x): I-bessel function of index nu and argument xbesseljbesselj(nu,x): J-bessel function of index nu and argument xbesseljhbesseljh(n,x): J-bessel function of index n+1/2 and argument x, where n is a non-negative integerbesselkbesselk(nu,x,{flag=0}): K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type). flag is optional, and may be set to 0: default; 1: use hyperubesselnbesseln(nu,x): N-bessel function of index nu and argument xbestappr(x,k): gives the best approximation to the real x with denominator less or equal to kbezoutbezout(x,y): gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dbezoutresbezoutres(x,y): gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbigomega(x): number of prime divisors of x, counted with multiplicitybinary(x): gives the vector formed by the binary digits of x (x integer)binomialbinomial(x,y): binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbitand(x,y): bitwise "and" of two integers x and y. Negative numbers behave as if modulo big power of 2GD-1,L,bitneg(x,{n=-1}): bitwise negation of an integers x truncated to n bits. n=-1 means represent infinite sequences of bit 1 as negative numbers. Negative numbers behave as if modulo big power of 2bitnegimplybitnegimply(x,y): bitwise "negated imply" of two integers x and y, in other words, x BITAND BITNEG(y). Negative numbers behave as if modulo big power of 2bitor(x,y): bitwise "or" of two integers x and y. Negative numbers behave as if modulo big power of 2bittestbittest(x,n): gives bit number n (coefficient of 2^n) of the integer x. Negative numbers behave as if modulo big power of 2bitxor(x,y): bitwise "exclusive or" of two integers x and y. Negative numbers behave as if modulo big power of 2bnfcertifylGbnfcertify(bnf): certify the correctness (i.e. remove the GRH) of the bnf data output by bnfclassunit or bnfinitbnfclassunitbnfclassunit(P,{flag=0},{tech=[]}): compute the class group, regulator of the number field defined by the polynomial P, and also the fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units. See manual for details about tech. P may also be a non-zero integer, and is then considered as the discriminant of a quadratic orderbnfclgpbnfclgp(P,{tech=[]}): compute the class group of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnfdecodemodulebnfdecodemodule(nf,fa): given a coded module fa as in bnrdisclist, gives the true modulebnfinit(P,{flag=0},{tech=[]}): compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units, 3: small bnfinit, which can be converted to a big one using bnfmake. See manual for details about techbnfisintnorm(bnf,x): compute a complete system of solutions (modulo units of positive norm) of the absolute norm equation N(a)=x, where a belongs to the maximal order of big number field bnf (if bnf is not certified, this depends on GRH)bnfisnormGGD1,L,pbnfisnorm(bnf,x,{flag=1}): Tries to tell whether x (in Q) is the norm of some fractional y (in bnf). Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-unit, with S a certain list of primes (in bnf) containing (among others) all primes dividing x. If bnf is known to be Galois, set flag=0 (in this case, x is a norm iff b=1). If flag is non zero the program adds to S all the primes : dividing flag if flag<0, or less than flag if flag>0. The answer is guaranteed (i.e x norm iff b=1) under GRH, if S contains all primes less than 12.log(disc(Bnf))^2, where Bnf is the Galois closure of bnfbnfisprincipalGGD1,L,bnfisprincipal(bnf,x,{flag=1}): bnf being output by bnfinit (with flag<=2), gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. flag is optional, whose meaning is: 0: output only v; 1: default; 2: output only v, precision being doubled until the result is obtained; 3: as 2 but output generatorsbnfissunitbnfissunit(bnf,sfu,x): bnf being output by bnfinit (with flag<=2), sfu by bnfsunit, gives the column vector of exponents of x on the fundamental S-units and the roots of unity if x is a unit, the empty vector otherwisebnfisunitbnfisunit(bnf,x): bnf being output by bnfinit (with flag<=2), gives the column vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisebnfmakebnfmake(sbnf): transforms small sbnf as output by bnfinit with flag=3 into a true big bnfbnfnarrowbnfnarrow(bnf): given a big number field as output by bnfinit, gives as a 3-component vector the structure of the narrow class groupbnfregbnfreg(P,{tech=[]}): compute the regulator of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnfsignunitbnfsignunit(bnf): matrix of signs of the real embeddings of the system of fundamental units found by bnfinitbnfsunitbnfsunit(bnf,S): compute the fundamental S-units of the number field bnf output by bnfinit, S being a list of prime ideals. res[1] contains the S-units, res[5] the S-classgroup. See manual for detailsbnfunitbnfunit(bnf): compute the fundamental units of the number field bnf output by bnfinit when they have not yet been computed (i.e. with flag=2)bnrL1(bnr, {subgroup}, {flag=0}): bnr being output by bnrinit(,,1) and subgroup being a square matrix defining a congruence subgroup of bnr (the trivial subgroup if omitted), for each character of bnr trivial on this subgroup, compute L(1, chi) (or equivalently the first non-zero term c(chi) of the expansion at s = 0). The binary digits of flag mean 1: if 0 then compute the term c(chi) and return [r(chi), c(chi)] where r(chi) is the order of L(s, chi) at s = 0, or if 1 then compute the value at s = 1 (and in this case, only for non-trivial characters), 2: if 0 then compute the value of the primitive L-function associated to chi, if 1 then compute the value of the L-function L_S(s, chi) where S is the set of places dividing the modulus of bnr (and the infinite places), 3: return also the charactersbnrclassbnrclass(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this module. flag is optional, and can be 0: default, 1: compute data necessary for working in the ray class group, for example with functions such as bnrisprincipal or bnrdisc, without computing the generators of the ray class group, or 2: with the generators. When flag=1 or 2, the fifth component is the ray class group structure obtained when flag=0bnrclassno(bnf,x): ray class number of the module x for the big number field bnf. Faster than bnrclass if only the ray class number is wantedbnrclassnolistbnrclassnolist(bnf,list): if list is as output by ideallist or similar, gives list of corresponding ray class numbersbnrconductorGDGDGDGbnrconductor(a1,{a2},{a3},{flag=0}): conductor f of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc). flag is optional and can be 0: default, 1: returns [f, Cl_f, H], H subgroup of the ray class group modulo f defining the extension, 2: returns [f, bnr(f), H]bnrconductorofcharbnrconductorofchar(bnr,chi): conductor of the character chi on the ray class group bnrbnrdiscGDGDGD0,L,bnrdisc(a1,{a2},{a3},{flag=0}): absolute or relative [N,R1,discf] of the field defined by a1,a2,a3. [a1,{a2},{a3}] is of type [bnr], [bnr,subgroup], [bnf, module] or [bnf,module,subgroup], where bnf is as output by bnfclassunit (with flag<=2), bnr by bnrclass (with flag>0), and subgroup is the HNF matrix of a subgroup of the corresponding ray class group (if omitted, the trivial subgroup). flag is optional whose binary digits mean 1: give relative data; 2: return 0 if module is not the conductorbnrdisclistbnrdisclist(bnf,bound,{arch}): gives list of discriminants of ray class fields of all conductors up to norm bound, in a long vector The ramified Archimedean places are given by arch; all possible values are taken if arch is omitted. Supports the alternative syntax bnrdisclist(bnf,list), where list is as output by ideallist or ideallistarch (with units)bnrinitbnrinit(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data linked to the ray class group structure corresponding to this module. flag is optional, and can be 0: default (same as bnrclass with flag = 1), 1: compute also the generators (same as bnrclass with flag = 2). The fifth component is the ray class group structurebnrisconductorlGDGDGbnrisconductor(a1,{a2},{a3}): returns 1 if the modulus is the conductor of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc), and 0 otherwise. Slightly faster than bnrconductor if this is the only desired resultbnrisprincipalbnrisprincipal(bnr,x,{flag=1}): bnr being output by bnrinit, gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. If (optional) flag is set to 0, output only vbnrrootnumber(bnr,chi,{flag=0}); returns the so-called Artin Root Number, i.e. the constant W appearing in the functional equation of the Hecke L-function associated to chi. Set flag = 1 if the character is known to be primitivebnrstark(bnr,{subgroup}): bnr being as output by bnrinit(,,1), finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the trivial subgroup if omitted) using Stark's units. The ground field and the class field must be totally real.break({n=1}): interrupt execution of current instruction sequence, and exit from the n innermost enclosing loopsceil(x): ceiling of x=smallest integer>=xcenterliftcenterlift(x,{v}): centered lift of x. Same as lift except for integermodschangevar(x,y): change variables of x according to the vector ycharpoly(A,{v=x},{flag=0}): det(v*Id-A)=characteristic polynomial of the matrix or polmod A. flag is optional and may be set to 1 (use Lagrange interpolation) or 2 (use Hessenberg form), 0 being the defaultchinese(x,{y}): x,y being both intmods (or polmods) computes z in the same residue classes as x and ycomponent(x,s): the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]. For list objects such as nf, bnf, bnr or ell, it is much easier to use member functions starting with "."concat(x,{y}): concatenation of x and y, which can be scalars, vectors or matrices, or lists (in this last case, both x and y have to be lists). If y is omitted, x has to be a list or row vector and its elements are concatenatedconj(x): the algebraic conjugate of xconjvec(x): conjugate vector of the algebraic number xcontent(x): gcd of all the components of x, when this makes sensecontfraccontfrac(x,{b},{lmax}): continued fraction expansion of x (x rational,real or rational function). b and lmax are both optional, where b is the vector of numerators of the continued fraction, and lmax is a bound for the number of terms in the continued fraction expansioncontfracpnqncontfracpnqn(x): [p_n,p_{n-1}; q_n,q_{n-1}] corresponding to the continued fraction xcorecore(n,{flag=0}): unique (positive of negative) squarefree integer d dividing n such that n/d is a square. If (optional) flag is non-null, output the two-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecoredisccoredisc(n,{flag=0}): discriminant of the quadratic field Q(sqrt(n)). If (optional) flag is non-null, output a two-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integercos(x): cosine of xcosh(x): hyperbolic cosine of xcotan(x): cotangent of xD"",r,D"",s,D0,L,default({opt},{v}): returns the current value of the current default opt. If v is present, set opt to v first. If no argument is given, print a list of all defaults as well as their values.denominatordenominator(x): denominator of x (or lowest common denominator in case of an array)derivderiv(x,{y}): derivative of x with respect to the main variable of y, or to the main variable of x if y is omitteddilogdilog(x): dilogarithm of xdirdiv(x,y): division of the Dirichlet series x by the Dirichlet series yV=GGEDGdireuler(p=a,b,expr,{c}): Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s). If c is present, output only the first c termsdirmuldirmul(x,y): multiplication of the Dirichlet series x by the Dirichlet series ydirzetak(nf,b): Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1divisors(x): gives a vector formed by the divisors of x in increasing orderdivrem(x,y,{v}): euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainder, with respect to v (to main variable if v is omitted)eint1(x,{n}): exponential integral E1(x). If n is present, computes the vector of the first n values of the exponential integral E1(n.x) (x > 0)elladdelladd(e,z1,z2): sum of the points z1 and z2 on elliptic curve eellak(e,n): computes the n-th Fourier coefficient of the L-function of the elliptic curve eellanellan(e,n): computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<2^24 on a 32-bit machine)ellapellap(e,p,{flag=0}): computes a_p for the elliptic curve e using Shanks-Mestre's method. flag is optional and can be set to 0 (default) or 1 (use Jacobi symbols)ellbilellbil(e,z1,z2): canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsellchangecurveellchangecurve(x,y): change data on elliptic curve according to y=[u,r,s,t]ellchangepointellchangepoint(x,y): change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]ellconvertname(name): convert an elliptic curve name (as found in the elldata database) from a string to a triplet [conductor, isogeny class, index]. It will also convert a triplet back to a curve name.elleisnum(om,k,{flag=0}): om=[om1,om2] being a 2-component vector giving a basis of a lattice L and k an even positive integer, computes the numerical value of the Eisenstein series of weight k. When flag is non-zero and k=4 or 6, this gives g2 or g3 with the correct normalizationelletaelleta(om): om=[om1,om2], returns the two-component row vector [eta1,eta2] of quasi-periods associated to [om1,om2]ellgeneratorsellgenerators(E): if E is an elliptic curve as output by ellinit(), return the generators of the Mordell-Weil group associated to the curve. This function depends on the curve being referenced in the elldata database.ellglobalredellglobalred(e): e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pellheightGGD2,L,pellheight(e,x,{flag=2}): canonical height of point x on elliptic curve E defined by the vector e. flag is optional and selects the algorithm used to compute the archimedean local height. Its meaning is 0: use theta-functions, 1: use Tate's method, 2: use Mestre's AGMellheightmatrixellheightmatrix(e,x): gives the height matrix for vector of points x on elliptic curve e using theta functionsellidentifyellidentify(E): look up the elliptic curve E in the elldata database and return [[N, M, ...], C] where N is the name of the curve in J. E. Cremona database, M the minimal model and C the coordinates change (see ellchangecurve).ellinit(x,{flag=0}): x being the vector [a1,a2,a3,a4,a6] defining the curve Y^2 + a1.XY + a3.Y = X^3 + a2.X^2 + a4.X + a6, gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,disc,j,[e1,e2,e3],w1,w2,eta1,eta2,area]. If the curve is defined over a p-adic field, the last six components are replaced by root,u^2,u,q,w,0. If optional flag is 1, omit them altogether. x can also be a string, in this case the coefficients of the curve with matching name are looked in the elldata database if available.ellisoncurveellisoncurve(e,x): true(1) if x is on elliptic curve e, false(0) if notelljellj(x): elliptic j invariant of xelllocalred(e,p): e being an elliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the Kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_pelllserieselllseries(e,s,{A=1}): L-series at s of the elliptic curve e, where A a cut-off point close to 1ellminimalmodelellminimalmodel(e,{&v}): return the standard minimal integral model of the rational elliptic curve e. Sets v to the corresponding change of variablesellorderellorder(e,p): order of the point p on the elliptic curve e over Q, 0 if non-torsionellordinateellordinate(e,x): y-coordinates corresponding to x-ordinate x on elliptic curve eellpointtozellpointtoz(e,P): lattice point z corresponding to the point P on the elliptic curve eellpow(e,x,n): n times the point x on elliptic curve e (n in Z)ellrootnolGDGellrootno(e,{p=1}): root number for the L-function of the elliptic curve e. p can be 1 (default), global root number, or a prime p (including 0) for the local root number at pellsearch(N): if N is an integer, it is taken as a conductor else if N is a string, it can be a curve name ("11a1"), a isogeny class ("11a") or a conductor ("11"). Return all curves in the elldata database that match the  property.ellsigmaellsigma(om,z,{flag=0}): om=[om1,om2], value of the Weierstrass sigma function of the lattice generated by om at z if flag = 0 (default). If flag = 1, arbitrary determination of the logarithm of sigma. If flag = 2 or 3, same but using the product expansion instead of theta seriesellsubellsub(e,z1,z2): difference of the points z1 and z2 on elliptic curve eelltaniyamaelltaniyama(e): modular parametrization of elliptic curve eelltorselltors(e,{flag=0}): torsion subgroup of elliptic curve e: order, structure, generators. If flag = 0, use Doud's algorithm; if flag = 1, use Lutz-NagellGDGD0,L,pPellwp(e,{z=x},{flag=0}): Complex value of Weierstrass P function at z on the lattice generated over Z by e=[om1,om2] (e as given by ellinit is also accepted). Optional flag means 0 (default), compute only P(z), 1 compute [P(z),P'(z)], 2 consider om as an elliptic curve and compute P(z) for that curve (identical to ellztopoint in that case). If z is omitted or is a simple variable, return formal expansion in zellzetaellzeta(om,z): om=[om1,om2], value of the Weierstrass zeta function of the lattice generated by om at zellztopointellztopoint(e,z): coordinates of point P on the curve e corresponding to the complex number zerfcerfc(x): complementary error functionvs*error("msg"): abort script with error message msgeta(x,{flag=0}): if flag=0, eta function without the q^(1/24), otherwise eta of the complex number x in the upper half plane intelligently computed using SL(2,Z) transformationseulerphieulerphi(x): Euler's totient function of xeval(x): evaluation of x, replacing variables by their valueexp(x): exponential of xfactor(x,{lim}): factorization of x. lim is optional and can be set whenever x is of (possibly recursive) rational type. If lim is set return partial factorization, using primes up to lim (up to primelimit if lim=0)factorback(f,{e},{nf}): given a factorisation f, gives the factored object back. If this is a prime ideal factorisation you must supply the corresponding number field as last argument. If e is present, f has to be a vector of the same length, and we return the product of the f[i]^e[i]factorcantorfactorcantor(x,p): factorization mod p of the polynomial x using Cantor-Zassenhausfactorff(x,p,a): factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factorialfactorial(x): factorial of x (x C-integer), the result being given as a real numberfactorint(x,{flag=0}): factor the integer x. flag is optional, whose binary digits mean 1: avoid MPQS, 2: avoid first-stage ECM (may fall back on it later), 4: avoid Pollard-Brent Rho and Shanks SQUFOF, 8: skip final ECM (huge composites will be declared prime)factormod(x,p,{flag=0}): factorization mod p of the polynomial x using Berlekamp. flag is optional, and can be 0: default or 1: simple factormod, same except that only the degrees of the irreducible factors are givenfactornf(x,t): factorization of the polynomial x over the number field defined by the polynomial tGGLD0,L,factorpadic(x,p,r,{flag=0}): p-adic factorization of the polynomial x to precision r. flag is optional and may be set to 0 (use round 4) or 1 (use Buchmann-Lenstra)GLDnffinit(p,n,{v=x}): monic irreducible polynomial of degree n over F_p[v]fibonaccifibonacci(x): fibonacci number of index x (x C-integer)floor(x): floor of x = largest integer<=xforvV=GGIfor(X=a,b,seq): the sequence is evaluated, X going from a up to bfordivvGVIfordiv(n,X,seq): the sequence is evaluated, X running over the divisors of nforellvVLLIforell(E,a,b,seq): execute seq for each elliptic curves E of conductor between a and b in the elldata database.forprimeforprime(X=a,b,seq): the sequence is evaluated, X running over the primes between a and bvV=GGGIforstep(X=a,b,s,seq): the sequence is evaluated, X going from a to b in steps of s (can be a vector of steps)vV=GDGIforsubgroup(H=G,{bound},seq): execute seq for each subgroup H of the abelian group G (in SNF form), whose index is bounded by bound. H is given as a left divisor of G in HNF formforvec(x=v,seq,{flag=0}): v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1 if flag is zero or omitted. If flag = 1 (resp. flag = 2), restrict to increasing (resp. strictly increasing) sequencesfrac(x): fractional part of x = x-floor(x)galoisexportgaloisexport(gal,{flag}): gal being a galois field as output by galoisinit, output a string representing the underlying permutation group in GAP notation (default) or Magma notation (flag = 1)GGD0,L,Dngaloisfixedfield(gal,perm,{flag},{v=y}): gal being a galois field as output by galoisinit and perm an element of gal.group or a vector of such elements, return [P,x] such that P is a polynomial defining the fixed field of gal[1] by the subgroup generated by perm, and x is a root of P in gal expressed as a polmod in gal.pol. If flag is 1 return only P. If flag is 2 return [P,x,F] where F is the factorization of gal.pol over the field defined by P, where the variable v stands for a root of Pgaloisidentifygaloisidentify(gal): gal being a galois field as output by galoisinit, output the isomorphism class of the underlying abstract group as a two-components vector [o,i], where o is the group order, and i is the group index in the GAP4 small group librarygaloisinit(pol,{den}): pol being a polynomial or a number field as output by nfinit defining a Galois extension of Q, compute the Galois group and all neccessary informations for computing fixed fields. den is optional and has the same meaning as in nfgaloisconj(,4)(see manual)galoisisabelian(gal,{flag=0}): gal being as output by galoisinit, return 0 if gal is not abelian, the HNF matrix of gal over gal.gen if flag=0, 1 if flag is 1, and the SNF of gal is flag=2galoispermtopolgaloispermtopol(gal,perm): gal being a galois field as output by galoisinit and perm a element of gal.group, return the polynomial defining the corresponding Galois automorphismGDGD0,L,Dngaloissubcyclo(N,H,{fl=0},{v}):Compute a polynomial (in variable v) defining the subfield of Q(zeta_n) fixed by the subgroup H of (Z/nZ)*. N can be an integer n, znstar(n) or bnrinit(bnfinit(y),[n,[1]],1). H can be given by a generator, a set of generator given by a vector or a HNF matrix (see manual). If flag is 1, output only the conductor of the abelian extension. If flag is 2 output [pol,f] where pol is the polynomial and f the conductor.galoissubfieldsgaloissubfields(G,{flags=0},{v}):Output all the subfields of G. flags have the same meaning as for galoisfixedfieldgaloissubgroupsgaloissubgroups(G):Output all the subgroups of Ggamma(x): gamma function at xgammahgammah(x): gamma of x+1/2 (x integer)gcd(x,{y}): greatest common divisor of x and y.getheapgetheap(): 2-component vector giving the current number of objects in the heap and the space they occupygetrandgetrand(): current value of random number seedgetstackgetstack(): current value of stack pointer avmagettime(): time (in milliseconds) since last call to gettimeglobalglobal(x): declare x to be a global variablelGGDGhilbert(x,y,{p}): Hilbert symbol at p of x,y.hyperu(a,b,x): U-confluent hypergeometric functionidealaddidealadd(nf,x,y): sum of two ideals x and y in the number field defined by nfidealaddtoone(nf,x,{y}): if y is omitted, when the sum of the ideals in the number field K defined by nf and given in the vector x is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1. Otherwise, x and y are ideals, and if they sum up to 1, find one element in each of them such that the sum is 1idealappridealappr(nf,x,{flag=0}): x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other p. If (optional) flag is non-null x must be a prime ideal factorization with possibly zero exponentsidealchinese(nf,x,y): x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealcoprimeidealcoprime(nf,x,y): gives an element b in nf such that b. x is an integral ideal coprime to the integral ideal yidealdivGGGD0,L,idealdiv(nf,x,y,{flag=0}): quotient x/y of two ideals x and y in HNF in the number field nf. If (optional) flag is non-null, the quotient is supposed to be an integral ideal (slightly faster)idealfactor(nf,x): factorization of the ideal x given in HNF into prime ideals in the number field nfidealhnf(nf,a,{b}): hermite normal form of the ideal a in the number field nf, whatever form a may have. If called as idealhnf(nf,a,b), the ideal is given as aZ_K+bZ_K in the number field K defined by nfidealintersectidealintersect(nf,x,y): intersection of two ideals x and y in the number field defined by nfidealinv(nf,x,{flag=0}): inverse of the ideal x in the number field nf. If flag is omitted or set to 0, use the different. If flag is 1 do not use itideallistGLD4,L,ideallist(nf,bound,{flag=4}): vector of vectors L of all idealstar of all ideals of norm<=bound. If (optional) flag is present, its binary digits are toggles meaning 1: give generators; 2: add units; 4: give only the ideals and not the bid.ideallistarchideallistarch(nf,list,arch): list is a vector of vectors of of bid's as output by ideallist. Return a vector of vectors with the same number of components as the original list. The leaves give information about moduli whose finite part is as in original list, in the same order, and archimedean part is now arch. The information contained is of the same kind as was present in the input.ideallog(nf,x,bid): if bid is a big ideal, as given by idealstar(nf,I,1) or idealstar(nf,I,2), gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)idealminidealmin(nf,ix,{vdir}): minimum of the ideal ix in the direction vdir in the number field nfidealmul(nf,x,y,{flag=0}): product of the two ideals x and y in the number field nf. If (optional) flag is non-nul, reduce the resultidealnormidealnorm(nf,x): norm of the ideal x in the number field nfidealpow(nf,x,n,{flag=0}): n-th power of the ideal x in HNF in the number field nf If (optional) flag is non-null, reduce the resultidealprimedec(nf,p): prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a. Z_K, e,f as usual, a as vector of components on the integral basis, b Lenstra's constantidealprincipalidealprincipal(nf,x): returns the principal ideal generated by the algebraic number x in the number field nfidealred(nf,x,{vdir=0}): LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealstar(nf,I,{flag=1}): gives the structure of (Z_K/I)^*. flag is optional, and can be 0: simply gives the structure as a 3-component vector v such that v[1] is the order (i.e. eulerphi(I)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generators. If flag=1 (default), gives idealstarinit, i.e. a 6-component vector [I,v,fa,f2,U,V] where v is as above without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*. Finally if flag=2, same as with flag=1 except that the generators are also givenidealtwoelt(nf,x,{a}): two-element representation of an ideal x in the number field nf. If (optional) a is non-zero, first element will be equal to aidealvallGGGidealval(nf,x,p): valuation at p given in idealprimedec format of the ideal x in the number field nfideleprincipalideleprincipal(nf,x): returns the principal idele generated by the algebraic number x in the number field nfifif(a,seq1,seq2): if a is nonzero, seq1 is evaluated, otherwise seq2. seq1 and seq2 are optional, and if seq2 is omitted, the preceding comma can be omitted alsoimag(x): imaginary part of xincgamincgam(s,x,{y}): incomplete gamma function. y is optional and is the precomputed value of gamma(s)incgamcincgamc(s,x): complementary incomplete gamma functionintcircV=GGEDGpintcirc(X=a,R,s,{tab}): numerical integration of s on the circle  |z-a|=R, divided by 2*I*Pi. tab is as in intnum.intformal(x,{y}): formal integration of x with respect to the main variable of y, or to the main variable of x if y is omittedintfouriercosV=GGGEDGpintfouriercos(X=a,b,x,s,{tab}): numerical integration from a to b of cos(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the cosine-Fourier transform if a=-infty and b=+infty.intfourierexpintfourierexp(X=a,b,x,s,{tab}): numerical integration from a to b of exp(-2*I*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the ordinary Fourier transform if a=-infty and b=+infty. Note the minus sign.intfouriersinintfouriersin(X=a,b,x,s,{tab}): numerical integration from a to b of sin(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the sine-Fourier transform if a=-infty and b=+infty.intfuncinitV=GGED0,L,D0,L,pintfuncinit(X=a,b,s,{flag=0},{m=0}): initialize tables for integrations  from a to b using a weight s(X). Essential for integral transforms such as intmellininv, intlaplaceinv and intfourier, since it avoids recomputing all the time the same quantities. Must then be used with intmellininvshort (for intmellininv) and directly with intnum and not with the corresponding  integral transforms for the others. See help for intnum for coding of a  and b, and m is as in intnuminit. If flag is nonzero, assumes that  s(-X)=conj(s(X)), which is twice faster.intlaplaceinvintlaplaceinv(X=sig,x,s,{tab}): numerical integration on the line real(z) = sig of s(z)exp(xz)dz/(2*I*Pi), i.e. inverse Laplace transform of s at x. tab is as in intnum.intmellininvintmellininv(X=sig,x,s,{tab}): numerical integration on the  line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin  transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). tab is as in intnum. Use  intmellininvshort if several values must be computed.intmellininvshortintmellininvshort(sig,x,tab): numerical integration on the  line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). Compulsory table tab has been  precomputed using the command intfuncinit(t=[[-1],sig[2]],[[1],sig[2]],s)  (with possibly its two optional additional parameters), where sig[2] = 1 if not given. Orders of magnitude faster than intmellininv.intnum(X=a,b,s,{tab}): numerical integration of s from a to b with  respect to X. a (and similarly b) is coded as follows. It can be a scalar: f is assumed to be C^infty at a. It can be a two component vector [a[1],a[2]], where a[1] is the scalar, and a[2] is the singularity exponent (in ]-1,0]), logs being neglected. It can be a one component vector [1] or [-1] meaning +infty or -infty, slowly decreasing functions. It can be a two component vector [[1], z] or [[-1], z], where [1] or [-1] indicates +infty or -infty and z is coded as follows. If z is zero, slowly decreasing. If z is real positive, exponentially decreasing, of the type exp(-zX). If z<-1, very slowly decreasing like X^(-z). If z is complex nonreal, real part is ignored and if z = r+I*s then if s>0, cosine oscillation exactly cos(sX), while if s<0, sine oscillation exactly sin(sX). If f is exponentially decreasing times oscillating function, you have a choice, but it is in general better to choose the oscillating part. Finally tab is either 0 (let the program choose  the integration step), a positive integer m (choose integration step 1/2^m), or a table tab precomputed with intnuminit (depending on the type of interval: compact, semi-compact or R, very slow, slow, exponential, or cosine or sine-oscillating decrease).intnuminit(a,b,{m=0}): initialize tables for integrations from a to b. See help for intnum for coding of a and b. Possible types: compact interval, semi-compact (one extremity at + or - infinity) or R, and very slowly, slowly or exponentially decreasing, or sine or cosine oscillating at infinities,VGGED0,L,D0,L,pintnuminitgen(t,a,b,ph,{m=0},{flag=0}): initialize tables for  integrations from a to b using abcissas ph(t) and weights ph'(t). Note that  there is no equal sign after the variable name t since t always goes from  -infty to +infty, but it is ph(t) which goes from a to b, and this is not  checked. If flag = 1 or 2, multiply the reserved table length by 4^flag, to  avoid corresponding error.intnumrombV=GGED0,L,pintnumromb(X=a,b,s,{flag=0}): numerical integration of s (smooth in  ]a,b[) from a to b with respect to X. flag is optional and mean 0: default.  s can be evaluated exactly on [a,b]; 1: general function; 2: a or b can be  plus or minus infinity (chosen suitably), but of same sign; 3: s has only  limits at a or bintnumstepintnumstep(): gives the default value of m used by all intnum and sumnum  routines, such that the integration step is 1/2^m.isfundamentalisfundamental(x): true(1) if x is a fundamental discriminant (including 1), false(0) if notispowerlGDGD&ispower(x,{k},{&n}): true (1) if x is a k-th power, false (0) if not. If n is given and a k-th root was computed in the process, put that in n. If k is omitted, return the maximal k >= 2 such that x = n^k is a perfect power, or 0 if no such k exist.isprime(x,{flag=0}): true(1) if x is a (proven) prime number, false(0) if not. If flag is 0 or omitted, use a combination of algorithms. If flag is 1, the primality is certified by the Pocklington-Lehmer Test. If flag is 2, the primality is certified using the APRCL test.ispseudoprimeispseudoprime(x,{n}): true(1) if x is a strong pseudoprime, false(0) if not. If n is 0 or omitted, use BPSW test, otherwise use strong Rabin-Miller test for n randomly chosen basesissquare(x,{&n}): true(1) if x is a square, false(0) if not. If n is given puts the exact square root there if it was computedissquarefree(x): true(1) if x is squarefree, false(0) if notvSkill(x): kills the present value of the variable or function x. Returns new value or 0kroneckerkronecker(x,y): kronecker symbol (x/y)lcmlcm(x,{y}): least common multiple of x and y, i.e. x*y / gcd(x,y)lengthlength(x): number of non code words in x, number of characters for a stringiGGlex(x,y): compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)lift(x,{v}): lifts every element of Z/nZ to Z or T[x]/PT[x] to T[x] for a type T if v is omitted, otherwise lift only polymods with main variable v. If v does not occur in x, lift only intmodslindep(x,{flag=0}): Z-linear dependencies between components of x. flag is optional, and can be 0: default, PSLQ; -1: using Hastad et al; -2: returns a non-trivial kernel vector (not integral in general); positive, and in that case should be between 0.5 and 1.0 times the accuracy in decimal digits of x, using a standard LLLlistcreate(n): creates an empty list of maximum length nlistinsert(list,x,n): insert x at index n in list, shifting the remaining elements to the rightlistkillvGlistkill(list): kills listlistput(list,x,{n}): sets n-th element of list equal to x. If n is omitted or greater than the current list length, just append xlistsortlistsort(list,{flag=0}): sort list in place. If flag is non-zero, suppress all but one occurence of each element in listlngamma(x): logarithm of the gamma function of xlog(x): natural logarithm of x.matadjointmatadjoint(x): adjoint matrix of xmatalgtobasis(nf,x): nfalgtobasis applied to every element of the matrix xmatbasistoalg(nf,x): nfbasistoalg applied to every element of the matrix xmatcompanionmatcompanion(x): companion matrix to polynomial xmatdetmatdet(x,{flag=0}): determinant of the matrix x using Gauss-Bareiss. If (optional) flag is set to 1, use classical gaussian elimination (slightly better for integer entries)matdetintmatdetint(x): some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with mathnfmodmatdiagonalmatdiagonal(x): creates the diagonal matrix whose diagonal entries are the entries of the vector xmateigenmateigen(x): eigenvectors of the matrix x given as columns of a matrixmatfrobenius(M,{flag},{v=x}): Return the Frobenius form of the square matrix M. If flag is 1, return only the elementary divisors as a vector of polynomials in the variable v. If flag is 2, return a two-components vector [F,B] where F is the Frobenius form and B is the basis change so that M=B^-1*F*B.mathessmathess(x): Hessenberg form of xmathilbertmathilbert(n): Hilbert matrix of order n (n C-integer)mathnfmathnf(A,{flag=0}): (upper triangular) Hermite normal form of A, basis for the lattice formed by the columns of A. flag is optional whose value range from 0 to 4 (0 if omitted), meaning : 0: naive algorithm. 1: Use Batut's algorithm. Output 2-component vector [H,U] such that H is the HNF of A, and U is a unimodular matrix such that AU=H. 3: Use Batut's algorithm. Output [H,U,P] where P is a permutation matrix such that P A U = H. 4: as 1, using a heuristic variant of LLL reduction along the waymathnfmod(x,d): (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is a multiple of the non-zero determinant of this latticemathnfmodidmathnfmodid(x,d): (upper triangular) Hermite normal form of x concatenated with d times the identity matrixmatidmatid(n): identity matrix of order n (n C-integer)matimage(x,{flag=0}): basis of the image of the matrix x. flag is optional and can be set to 0 or 1, corresponding to two different algorithmsmatimagecomplmatimagecompl(x): vector of column indices not corresponding to the indices given by the function matimagematindexrankmatindexrank(x): gives two extraction vectors (rows and columns) for the matrix x such that the extracted matrix is square of maximal rankmatintersectmatintersect(x,y): intersection of the vector spaces whose bases are the columns of x and ymatinverseimagematinverseimage(x,y): an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwisematisdiagonalmatisdiagonal(x): true(1) if x is a diagonal matrix, false(0) otherwisematkermatker(x,{flag=0}): basis of the kernel of the matrix x. flag is optional, and may be set to 0: default; non-zero: x is known to have integral entriesmatkerintmatkerint(x,{flag=0}): LLL-reduced Z-basis of the kernel of the matrix x with integral entries. flag is optional, and may be set to 0: default, uses a modified LLL, 1: uses matrixqzmatmuldiagonal(x,d): product of matrix x by diagonal matrix whose diagonal coefficients are those of the vector d, equivalent but faster than x*matdiagonal(d)matmultodiagonalmatmultodiagonal(x,y): product of matrices x and y, knowing that the result will be a diagonal matrix. Much faster than general multiplication in that casematpascalLDGmatpascal(n,{q}): Pascal triangle of order n if q is omited. q-Pascal triangle otherwisematrankmatrank(x): rank of the matrix xGGDVDVDImatrix(m,n,{X},{Y},{expr=0}): mXn matrix of expression expr, the row variable X going from 1 to m and the column variable Y going from 1 to n. By default, fill with 0smatrixqz(x,p): if p>=0, transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwise. If p=-1, finds a basis of the intersection with Z^n of the lattice spanned by the columns of x. If p=-2, finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsizematsize(x): number of rows and columns of the vector/matrix x as a 2-vectormatsnfmatsnf(x,{flag=0}): Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vector d. Binary digits of flag mean 1: returns [u,v,d] where d=u*x*v, otherwise only the diagonal d is returned, 2: allow polynomial entries, otherwise assume x is integral, 4: removes all information corresponding to entries equal to 1 in dmatsolvematsolve(M,B): gaussian solution of MX=B (M matrix, B column vector)matsolvemodmatsolvemod(M,D,B,{flag=0}): one solution of system of congruences MX=B mod D (M matrix, B and D column vectors). If (optional) flag is non-null return all solutionsmatsupplementmatsupplement(x): supplement the columns of the matrix x to an invertible matrixmattransposemattranspose(x): x~=transpose of xmax(x,y): maximum of x and ymin(x,y): minimum of x and yminpolyminpoly(A,{v=x}): minimal polynomial of the matrix or polmod A.modreverse(x): reverse polymod of the polymod x, if it existsmoebius(x): Moebius function of xnewtonpolynewtonpoly(x,p): Newton polygon of polynomial x with respect to the prime pnextnext({n=1}): interrupt execution of current instruction sequence, and start another iteration from the n-th innermost enclosing loopsnextprimenextprime(x): smallest pseudoprime >= xnfalgtobasis(nf,x): transforms the algebraic number x into a column vector on the integral basis nf.zkGD0,L,DGnfbasis(x,{flag=0},{p}): integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithm. Second and third args are optional. Binary digits of flag mean 1: assume that no square of a prime>primelimit divides the discriminant of x, 2: use round 2 algorithm instead. If present, p provides the matrix of a partial factorization of the discriminant of x, useful if one wants only an order maximal at certain primes onlynfbasistoalg(nf,x): transforms the column vector x on the integral basis into an algebraic numbernfdetintnfdetint(nf,x): multiple of the ideal determinant of the pseudo generating set xnfdisc(x,{flag=0},{p}): discriminant of the number field defined by the polynomial x using round 4. Optional args flag and p are as in nfbasisnfeltdivnfeltdiv(nf,a,b): element a/b in nfnfeltdiveucnfeltdiveuc(nf,a,b): gives algebraic integer q such that a-bq is smallnfeltdivmodprnfeltdivmodpr(nf,a,b,pr): element a/b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltdivremnfeltdivrem(nf,a,b): gives [q,r] such that r=a-bq is smallnfeltmodnfeltmod(nf,a,b): gives r such that r=a-bq is small with q algebraic integernfeltmulnfeltmul(nf,a,b): element a. b in nfnfeltmulmodprnfeltmulmodpr(nf,a,b,pr): element a. b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltpownfeltpow(nf,a,k): element a^k in nfnfeltpowmodprnfeltpowmodpr(nf,a,k,pr): element a^k modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltreducenfeltreduce(nf,a,id): gives r such that a-r is in the ideal id and r is smallnfeltreducemodprnfeltreducemodpr(nf,a,pr): element a modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltvalnfeltval(nf,a,pr): valuation of element a at the prime pr as output by idealprimedecnffactor(nf,x): factor polynomial x in number field nfnffactormod(nf,pol,pr): factorize polynomial pol modulo prime ideal pr in number field nfnfgaloisapplynfgaloisapply(nf,aut,x): Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfnfgaloisconj(nf,{flag=0},{den}): list of conjugates of a root of the polynomial x=nf.pol in the same number field. flag is optional (set to 0 by default), meaning 0: use combination of flag 4 and 1, always complete; 1: use nfroots; 2 : use complex numbers, LLL on integral basis (not always complete); 4: use Allombert's algorithm, complete if the field is Galois of degree <= 35 (see manual for detail). nf can be simply a polynomial with flag 0,2 and 4, meaning: 0: use combination of flag 4 and 2, not always complete (but a warning is issued when the list is not proven complete); 2 & 4: same meaning and restrictions. Note that only flag 4 can be applied to fields of large degrees (approx. >= 20)lGGGDGnfhilbert(nf,a,b,{p}): if p is omitted, global Hilbert symbol (a,b) in nf, that is 1 if X^2-aY^2-bZ^2 has a non-trivial solution (X,Y,Z) in nf, -1 otherwise. Otherwise compute the local symbol modulo the prime ideal pnfhnfnfhnf(nf,x): if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhnfmodnfhnfmod(nf,x,detx): if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfinit(pol,{flag=0}): pol being a nonconstant irreducible polynomial, gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]. flag is optional and can be set to 0: default; 1: do not compute different; 2: first use polred to find a simpler polynomial; 3: outputs a two-element vector [nf,Mod(a,P)], where nf is as in 2 and Mod(a,P) is a polymod equal to Mod(x,pol) and P=nf.pol; 4: as 2 but use a partial polred; 5: is to 3 what 4 is to 2nfisidealnfisideal(nf,x): true(1) if x is an ideal in the number field nf, false(0) if notnfisinclnfisincl(x,y): tests whether the number field x is isomorphic to a subfield of y (where x and y are either polynomials or number fields as output by nfinit). Return 0 if not, and otherwise all the isomorphisms. If y is a number field, a faster algorithm is usednfisisomnfisisom(x,y): as nfisincl but tests whether x is isomorphic to ynfkermodprnfkermodpr(nf,x,pr): kernel of the matrix x in Z_K/pr, where pr is in modpr format (see nfmodprinit)nfmodprinitnfmodprinit(nf,pr): transform the 5 element row vector pr representing a prime ideal into modpr format necessary for all operations mod pr in the number field nf (see manual for details about the format)nfnewprec(nf): transform the number field data nf into new data using the current (usually larger) precisionDGGnfroots({nf},pol): roots of polynomial pol belonging to nf (Q if omitted) without multiplicitynfrootsof1nfrootsof1(nf): number of roots of unity and primitive root of unity in the number field nfnfsnfnfsnf(nf,x): if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfsolvemodprnfsolvemodpr(nf,a,b,pr): solution of a*x=b in Z_K/pr, where a is a matrix and b a column vector, and where pr is in modpr format (see nfmodprinit)nfsubfields(nf,{d=0}): find all subfields of degree d of number field nf (all subfields if d is null or omitted). Result is a vector of subfields, each being given by [g,h], where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nfnorm(x): norm of xnorml2(x): square of the L2-norm of the vector xnumbpartnumbpart(x): number of partitions of xnumdiv(x): number of divisors of xnumeratornumerator(x): numerator of xnumtoperm(n,k): permutation number k (mod n!) of n letters (n C-integer)omega(x): number of distinct prime divisors of xpadicapprpadicappr(x,a): p-adic roots of the polynomial x congruent to a mod ppadicprec(x,p): absolute p-adic precision of object xpermtonum(vect): ordinal (between 1 and n!) of permutation vectpolcoeffpolcoeff(x,s,{v}): coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[]). With respect to the main variable if v is omitted, with respect to the variable v otherwisepolcompositumpolcompositum(pol1,pol2,{flag=0}): vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2. If (optional) flag is set (i.e non-null), output for each compositum, not only the compositum polynomial pol, but a vector [pol,al1,al2,k] where al1 (resp. al2) is a root of pol1 (resp. pol2) expressed as a polynomial modulo pol, and a small integer k such that al2+k*al1 is the chosen root of polpolcyclo(n,{v=x}): n-th cyclotomic polynomial (in variable v)poldegreelGDnpoldegree(x,{v}): degree of the polynomial or rational function x with respect to main variable if v is omitted, with respect to v otherwise. For scalar x, return 0 is x is non-zero and a negative number otherwisepoldiscpoldisc(x,{v}): discriminant of the polynomial x, with respect to main variable if v is omitted, with respect to v otherwisepoldiscreduced(f): vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fpolgaloispolgalois(x): Galois group of the polynomial x (see manual for group coding). Return [n, s, k, name] where n is the order, s the signature, k the index and name is the GAP4 name of the transitive group.GGGLpolhensellift(x, y, p, e): lift the factorization y of x modulo p to a factorization modulo p^e using Hensel lift. The factors in y must be pairwise relatively prime modulo pGDGDGD&polinterpolate(xa,{ya},{x},{&e}): polynomial interpolation at x according to data vectors xa, ya (ie return P such that P(xa[i]) = ya[i] for all i). If ya is omitter, return P such that P(i) = xa[i]. If present, e will contain an error estimate on the returned valuepolisirreduciblepolisirreducible(x): true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantpolleadpollead(x,{v}): leading coefficient of polynomial or series x, or x itself if x is a scalar. Error otherwise. With respect to the main variable of x if v is omitted, with respect to the variable v otherwisepollegendrepollegendre(n,{v=x}): legendre polynomial of degree n (n C-integer), in variable vpolrecippolrecip(x): reciprocal polynomial of xpolred(x,{flag=0},{p}): reduction of the polynomial x (gives minimal polynomials only). Second and third args are optional. The following binary digits of flag are significant 1: partial reduction, 2: gives also elements. p, if present, contains the complete factorization matrix of the discriminantpolredabs(x,{flag=0}): a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 norm. flag is optional, whose binary digit mean 1: give the element whose characteristic polynomial is the given polynomial. 4: give all polynomials of minimal T2 norm (give only one of P(x) and P(-x)). 16: partial reductionpolredordpolredord(x): reduction of the polynomial x, staying in the same orderpolresultantGGDnD0,L,polresultant(x,y,{v},{flag=0}): resultant of the polynomials x and y, with respect to the main variables of x and y if v is omitted, with respect to the variable v otherwise. flag is optional, and can be 0: default, assumes that the polynomials have exact entries (uses the subresultant algorithm), 1 for arbitrary polynomials, using Sylvester's matrix, or 2: using a Ducos's modified subresultant algorithmpolrootspolroots(x,{flag=0}): complex roots of the polynomial x. flag is optional, and can be 0: default, uses Schonhage's method modified by Gourdon, or 1: uses a modified Newton methodpolrootsmod(x,p,{flag=0}): roots mod p of the polynomial x. flag is optional, and can be 0: default, or 1: use a naive search, useful for small ppolrootspadicpolrootspadic(x,p,r): p-adic roots of the polynomial x to precision rpolsturmpolsturm(x,{a},{b}): number of real roots of the polynomial x in the interval]a,b] (which are respectively taken to be -oo or +oo when omitted)polsubcycloLLDnpolsubcyclo(n,d,{v=x}): finds an equation (in variable v) for the d-th degree subfields of Q(zeta_n). Output is a polynomial or a vector of polynomials is there are several such fields, or none.polsylvestermatrixpolsylvestermatrix(x,y): forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowspolsympolsym(x,n): vector of symmetric powers of the roots of x up to npoltchebipoltchebi(n,{v=x}): Tchebitcheff polynomial of degree n (n C-integer), in variable vpoltschirnhauspoltschirnhaus(x): random Tschirnhausen transformation of the polynomial xLGD0,L,ppolylog(m,x,{flag=0}): m-th polylogarithm of x. flag is optional, and can be 0: default, 1: D_m~-modified m-th polylog of x, 2: D_m-modified m-th polylog of x, 3: P_m-modified m-th polylog of xpolzagierpolzagier(n,m): Zagier's polynomials of index n,mprecision(x,{n}): change the precision of x to be n (n C-integer). If n is omitted, output real precision of object xprecprimeprecprime(x): largest pseudoprime <= x, 0 if x<=1prime(n): returns the n-th prime (n C-integer)primepi(x): the prime counting function pi(x) = #{p <= x, p prime}.primes(n): returns the vector of the first n primes (n C-integer)print(a): outputs a (in raw format) ending with newlineprint1print1(a): outputs a (in raw format) without ending with newlineprintpprintp(a): outputs a (in beautified format) ending with newlineprintp1printp1(a): outputs a (in beautified format) without ending with newlineprinttexprinttex(a): outputs a in TeX formatprodprod(X=a,b,expr,{x=1}): x times the product (X runs from a to b) of expressionprodeulerV=GGEpprodeuler(X=a,b,expr): Euler product (X runs over the primes between a and b) of real or complex expressionV=GED0,L,pprodinf(X=a,expr,{flag=0}): infinite product (X goes from a to infinity) of real or complex expression. flag can be 0 (default) or 1, in which case compute the product of the 1+expr insteadpsi(x): psi-function at xqfbclassnoqfbclassno(x,{flag=0}): class number of discriminant x using Shanks's method by default. If (optional) flag is set to 1, use Euler productsqfbcomprawqfbcompraw(x,y): Gaussian composition without reduction of the binary quadratic forms x and yqfbhclassnoqfbhclassno(x): Hurwitz-Kronecker class number of x>0qfbnucompqfbnucomp(x,y,l): composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputedqfbnupowqfbnupow(x,n): n-th power of primitive positive definite quadratic form x using nucomp and nuduplqfbpowrawqfbpowraw(x,n): n-th power without reduction of the binary quadratic form xqfbprimeformqfbprimeform(x,p): returns the prime form of discriminant x, whose first coefficient is pqfbredGD0,L,DGDGDGqfbred(x,{flag=0},{D},{isqrtD},{sqrtD}): reduction of the binary quadratic form x. All other args. are optional. D, isqrtD and sqrtD, if present, supply the values of the discriminant, floor(sqrt(D)) and sqrt(D) respectively. If D<0, its value is not used and all references to Shanks's distance hereafter are meaningless. flag can be any of 0: default, uses Shanks's distance function d; 1: use d, do a single reduction step; 2: do not use d; 3: do not use d, single reduction step.qfbsolveqfbsolve(Q,p): Return [x,y] so that Q(x,y)=p where Q is a binary quadratic form and p a prime number, or 0 if there is no solution.qfgaussredqfgaussred(x): square reduction of the (symmetric) matrix x (returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)qfjacobiqfjacobi(x): eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xqflllqflll(x,{flag=0}): LLL reduction of the vectors forming the matrix x (gives the unimodular transformation matrix). The columns of x must be linearly independent, unless specified otherwise below. flag is optional,  and can be 0: default, 1: assumes x is integral, columns may be dependent, 2: assumes x is integral, returns a partially reduced basis, 4: assumes x is  integral, returns [K,I] where K is the integer kernel of x and I the LLL reduced image, 5: same as 4 but x may have polynomial coefficients, 8: same as 0 but x may have polynomial coefficientsqflllgramqflllgram(x,{flag=0}): LLL reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix). flag is optional and can be 0: default,1: lllgramint algorithm for integer matrices, 4: lllgramkerim giving the kernel and the LLL reduced image, 5: lllgramkerimgen same when the matrix has polynomial coefficients, 8: lllgramgen, same as qflllgram when the coefficients are polynomialsqfminimqfminim(x,{bound},{maxnum},{flag=0}): number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0. flag is optional, and can be 0: default; 1: returns the first minimal vector found (ignore maxnum); 2: as 0 but uses a more robust, slower implementation, valid for non integral quadratic formsqfperfectionqfperfection(a): rank of matrix of xx~ for x minimal vectors of a gram matrix aqfrepqfrep(x,B,{flag=0}): vector of (half) the number of vectors of norms from 1 to B for the integral and definite quadratic form x. Binary digits of flag mean 1: count vectors of even norm from 1 to 2B, 2: return a t_VECSMALL instead of a t_VECqfsignqfsign(x): signature of the symmetric matrix xquadclassunit(D,{flag=0},{tech=[]}): compute the structure of the class group and the regulator of the quadratic field of discriminant D. If flag is non-null (and D>0), compute the narrow class group. See manual for the optional technical parametersquaddiscquaddisc(x): discriminant of the quadratic field Q(sqrt(x))quadgenquadgen(x): standard generator of quadratic order of discriminant xquadhilbert(D,{pq}): relative equation for the Hilbert class field of the quadratic field of discriminant D (which can also be a bnf). If D<0, pq (if supplied) is a 2-component vector [p,q], where p,q are the prime numbers needed for Schertz's method. In that case, return 0 if [p,q] not suitable.quadpolyquadpoly(D,{v=x}): quadratic polynomial corresponding to the discriminant D, in variable vquadray(D,f,{lambda}): relative equation for the ray class field of conductor f for the quadratic field of discriminant D (which can also be a bnf). For D < 0, lambda (if supplied) is the technical element of bnf  necessary for Schertz's method. In that case, return 0 if lambda is not suitable.quadregulatorquadregulator(x): regulator of the real quadratic field of discriminant xquadunitquadunit(x): fundamental unit of the quadratic field of discriminant x where x must be positiverandom({N=2^31}): random integer between 0 and N-1readvecreadvec({filename}): create a vector whose components are the evaluation of all the expressions found in the input file filenamereal(x): real part of xremoveprimesremoveprimes({x=[]}): remove primes in the vector x (with at most 100 components) from the prime table. x can also be a single integer. List the current extra primes if x is omittedreorder({x=[]}): reorder the variables for output according to the vector x. If x is void or omitted, print the current list of variablesreturnreturn({x=0}): return from current subroutine with result xrnfalgtobasis(rnf,x): relative version of nfalgtobasis, where rnf is a relative numberfieldrnfbasisrnfbasis(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setrnfbasistoalgrnfbasistoalg(rnf,x): relative version of nfbasistoalg, where rnf is a relative numberfieldGGGDnrnfcharpoly(nf,T,alpha,{var=x}): characteristic polynomial of alpha over nf, where alpha belongs to the algebra defined by T over nf. Returns a polynomial in variable var (x by default)rnfconductorrnfconductor(bnf,polrel,{flag=0}): conductor of the Abelian extension of bnf defined by polrel. The result is [conductor,rayclassgroup,subgroup], where conductor is the conductor itself, rayclassgroup the structure of the corresponding full ray class group, and subgroup the HNF defining the norm group (Artin or Takagi group) on the given generators rayclassgroup[3]. If flag is non-zero, check that polrel indeed defines an Abelian extensionrnfdedekindrnfdedekind(nf,T,pr): relative Dedekind criterion over nf, applied to the order defined by a root of irreducible polynomial T, modulo the prime ideal pr. Returns [flag,basis,val], where basis is a pseudo-basis of the enlarged order, flag is 1 iff this order is pr-maximal, and val is the valuation in pr of the order discriminantrnfdet(nf,order): given a pseudomatrix, compute its pseudodeterminantrnfdiscrnfdisc(nf,pol): given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfeltabstorelrnfeltabstorel(rnf,x): transforms the element x from absolute to relative representationrnfeltdownrnfeltdown(rnf,x): expresses x on the base field if possible; returns an error otherwisernfeltreltoabsrnfeltreltoabs(rnf,x): transforms the element x from relative to absolute representationrnfeltuprnfeltup(rnf,x): expresses x (belonging to the base field) on the relative fieldrnfequation(nf,pol,{flag=0}): given a pol with coefficients in nf, gives the absolute equation apol of the number field defined by pol. flag is optional, and can be 0: default, or non-zero, gives [apol,th], where th expresses the root of nf.pol in terms of the root of apolrnfhnfbasisrnfhnfbasis(bnf,order): given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfidealabstorelrnfidealabstorel(rnf,x): transforms the ideal x from absolute to relative representationrnfidealdownrnfidealdown(rnf,x): finds the intersection of the ideal x with the base fieldrnfidealhnfrnfidealhnf(rnf,x): relative version of idealhnf, where rnf is a relative numberfieldrnfidealmulrnfidealmul(rnf,x,y): relative version of idealmul, where rnf is a relative numberfieldrnfidealnormabsrnfidealnormabs(rnf,x): absolute norm of the ideal xrnfidealnormrelrnfidealnormrel(rnf,x): relative norm of the ideal xrnfidealreltoabsrnfidealreltoabs(rnf,x): transforms the ideal x from relative to absolute representationrnfidealtwoeltrnfidealtwoelt(rnf,x): relative version of idealtwoelt, where rnf is a relative numberfieldrnfidealuprnfidealup(rnf,x): lifts the ideal x (of the base field) to the relative fieldrnfinitrnfinit(nf,pol): pol being a non constant irreducible polynomial defined over the number field nf, initializes a vector of data necessary for working in relative number fields (rnf functions). See manual for technical detailsrnfisfreernfisfree(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notrnfisnormrnfisnorm(T,x,{flag=0}): T is as output by rnfisnorminit applied to L/K. Tries to tell whether x is a norm from L/K. Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-integer, with S a list of places in K containing the ramified primes, generators of the class group of ext, as well as those primes dividing x. If L/K is Galois, omit flag, otherwise it is used to add more places to S: all the places above the primes p <= flag (resp. p | flag) if flag > 0 (resp. flag < 0). The answer is guaranteed (i.e x norm iff b=1) if L/K is Galois or, under GRH, if S contains all primes less than 12.log(disc(M))^2, where M is the normal closure of L/KGGD2,L,rnfisnorminit(pol,polrel,{flag=2}): let K be defined by a root of pol, L/K the extension defined by polrel. Compute technical data needed by rnfisnorm to solve norm equations Nx = a, for x in L, and a in K. If flag=0, do not care whether L/K is Galois or not; if flag = 1, assume L/K is Galois; if flag = 2, determine whether L/K is Galoisrnfkummerrnfkummer(bnr,{subgroup},{deg=0}): bnr being as output by bnrinit, finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the ray class field if subgroup is omitted). deg can be zero (default), or positive, and in this case the output is the list of all relative equations of degree deg for the given bnrrnflllgramrnflllgram(nf,pol,order): given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixrnfnormgroup(bnr,polrel): norm group (or Artin or Takagi group) corresponding to the Abelian extension of bnr.bnf defined by polrel, where the module corresponding to bnr is assumed to be a multiple of the conductor. The result is the HNF defining the norm group on the given generators in bnr[5][3]rnfpolredrnfpolred(nf,pol): given a pol with coefficients in nf, finds a list of relative polynomials defining some subfields, hopefully simplerrnfpolredabs(nf,pol,{flag=0}): given a pol with coefficients in nf, finds a relative simpler polynomial defining the same field. Binary digits of flag mean: 1: return also the element whose characteristic polynomial is the given polynomial, 2: return an absolute polynomial, 16: partial reductionrnfpseudobasisrnfpseudobasis(nf,pol): given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfsteinitzrnfsteinitz(nf,order): given an order as output by rnfpseudobasis, gives [A,I,D,d] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialround(x,{&e}): take the nearest integer to all the coefficients of x. If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitsserconvolserconvol(x,y): convolution (or Hadamard product) of two power seriesserlaplaceserlaplace(x): replaces the power series sum of a_n*x^n/n! by sum of a_n*x^n. For the reverse operation, use serconvol(x,exp(X))serreverse(x): reversion of the power series xsetintersect(x,y): intersection of the sets x and ysetissetsetisset(x): true(1) if x is a set (row vector with strictly increasing entries), false(0) if notsetminus(x,y): set of elements of x not belonging to ysetrandsetrand(n): reset the seed of the random number generator to nlGGD0,L,setsearch(x,y,{flag=0}): looks if y belongs to the set x. If flag is 0 or omitted, returns 0 if it is not, otherwise returns the index j such that y==x[j]. If flag is non-zero, return 0 if y belongs to x, otherwise the index j where it should be insertedsetunion(x,y): union of the sets x and yshift(x,n): shift x left n bits if n>=0, right -n bits if n<0.shiftmulshiftmul(x,n): multiply x by 2^n (n>=0 or n<0)sigma(x,{k=1}): sum of the k-th powers of the divisors of x. k is optional and if omitted is assumed to be equal to 1iGsign(x): sign of x, of type integer, real or fractionsimplifysimplify(x): simplify the object x as much as possiblesin(x): sine of xsinh(x): hyperbolic sine of xsizebytesizebyte(x): number of bytes occupied by the complete tree of the object xsizedigitsizedigit(x): maximum number of decimal digits minus one of (the coefficients of) xsolve(X=a,b,expr): real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sqr(x): square of x. NOT identical to x*xsqrt(x): square root of xsqrtint(x): integer square root of x (x integer)GGD&psqrtn(x,n,{&z}): nth-root of x, n must be integer. If present, z is set to a suitable root of unity to recover all solutions. If it was not possible, z is set to zerosubgrouplist(bnr,{bound},{flag=0}): bnr being as output by bnrinit or a list of cyclic components of a finite Abelian group G, outputs the list of subgroups of G (of index bounded by bound, if not omitted), given as HNF left divisors of the SNF matrix corresponding to G. If flag=0 (default) and bnr is as output by bnrinit, gives only the subgroups for which the modulus is the conductorGnGsubst(x,y,z): in expression x, replace the variable y by the expression zsubstpolsubstpol(x,y,z): in expression x, replace the polynomial y by the expression z, using remainder decomposition of x.substvec(x,v,w): in expression x, make a best effort to replace the variables v1,...,vn by the expression w1,...,wnsum(X=a,b,expr,{x=0}): x plus the sum (X goes from a to b) of expression exprsumalt(X=a,expr,{flag=0}): Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialsGVEsumdiv(n,X,expr): sum of expression expr, X running over the divisors of nsuminf(X=a,expr): infinite sum (X goes from a to infinity) of real or complex expression exprV=GGEDGD0,L,psumnum(X=a,sig,expr,{tab},{flag=0}): numerical summation of expr from  X = ceiling(a) to +infinity. sig is either a scalar or a two-component vector coding the function's decrease rate at infinity. It is assumed that the scalar part of sig is to the right of all poles of expr. If present, tab must be initialized by sumnuminit. If flag is nonzero, assumes that conj(expr(z)) = expr(conj(z)).sumnumaltsumnumalt(X=a,sig,s,{tab},{flag=0}): numerical summation of (-1)^X s from X = ceiling(a) to +infinity. Note that the (-1)^X must not be included. sig is either a scalar or a two-component vector coded as in intnum, and the  scalar part is larger than all the real parts of the poles of s. Uses intnum, hence tab is as in intnum. If flag is nonzero, assumes that the function to  be summed satisfies conj(f(z))=f(conj(z)), and then up to twice faster.sumnuminitGD0,L,D1,L,psumnuminit(sig, {m=0}, {sgn=1}): initialize tables for numerical summation. sgn is 1 (in fact >= 0), the default, for sumnum (ordinary sums)  or -1 (in fact < 0) for sumnumalt (alternating sums). sig is as in sumnum and m is as in intnuminit.sumpos(X=a,expr,{flag=0}): sum of positive series expr, the formal variable X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialstan(x): tangent of xtanh(x): hyperbolic tangent of xtaylorGnPtaylor(x,y): taylor expansion of x with respect to the main variable of yteichmuller(x): teichmuller character of p-adic number xtheta(q,z): Jacobi sine theta-functionGLpthetanullk(q,k): k'th derivative at z=0 of theta(q,z)thue(tnf,a,{sol}): solve the equation P(x,y)=a, where tnf was created with thueinit(P), and sol, if present, contains the solutions of Norm(x)=a modulo units in the number field defined by P. If tnf was computed without assuming GRH (flag 1 in thueinit), the result is unconditionalthueinitthueinit(P,{flag=0}): initialize the tnf corresponding to P, that will be used to solve Thue equations P(x,y) = some-integer. If flag is non-zero, certify the result unconditionnaly. Otherwise, assume GRH (much faster of course)trace(x): trace of xtrapD"",r,DIDItrap({err}, {rec}, {seq}): try to execute seq, trapping error err (all of them if err ommitted); sequence rec is executed if the error occurs and is the result of the command. When seq is omitted, define rec as a default handler for error err (a break loop will be started if rec omitted). If rec is the empty string "" pop out the last default handlertruncatetruncate(x,{&e}): truncation of x; when x is a power series,take away the O(X^). If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitstype(x): return the type of the GEN x.untiluntil(a,seq): evaluate the expression sequence seq until a is nonzerovaluationvaluation(x,p): valuation of x with respect to pvariable(x): main variable of object x. Gives p for p-adic x, error for scalarsvecextract(x,y,{z}): extraction of the components of the matrix or vector x according to y and z. If z is omitted, y designs columns, otherwise y corresponds to rows and z to columns. y and z can be vectors (of indices), strings (indicating ranges as in "1..10") or masks (integers whose binary representation indicates the indices to extract, from left to right 1, 2, 4, 8, etc.)vecmax(x): maximum of the elements of the vector/matrix xvecmin(x): minimum of the elements of the vector/matrix xvecsort(x,{k},{flag=0}): sorts the vector of vectors (or matrix) x in ascending order, according to the value of its k-th component if k is not omitted. Binary digits of flag (if present) mean: 1: indirect sorting, return the permutation instead of the permuted vector, 2: sort using lexicographic order, 4: use descending instead of ascending ordervector(n,{X},{expr=0}): row vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0svectorsmallvectorsmall(n,{X},{expr=0}): VECSMALL with n components of expression expr (X ranges from 1 to n) which must be small integers. By default, fill with 0svectorvvectorv(n,{X},{expr=0}): column vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0sweberweber(x,{flag=0}): One of Weber's f function of x. flag is optional, and can be 0: default, function f(x)=exp(-i*Pi/24)*eta((x+1)/2)/eta(x) such that (j=(f^24-16)^3/f^24), 1: function f1(x)=eta(x/2)/eta(x) such that (j=(f1^24+16)^3/f2^24), 2: function f2(x)=sqrt(2)*eta(2*x)/eta(x) such that (j=(f2^24+16)^3/f2^24)whilewhile(a,seq): while a is nonzero evaluate the expression sequence seq. Otherwise 0writevss*write(filename,a): write the string expression a (same output as print) to filenamewrite1write1(filename,a): write the string expression a (same output as print1) to filenamevsDGwritebin(filename,{x}): write x as a binary object to file filename. If x is omitted, write all session variableswritetexwritetex(filename,a): write the string expression a (same format as print) to filename, in TeX formatzeta(s): Riemann zeta function at s with s a complex or a p-adic numberzetak(nfz,s,{flag=0}): Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by zetakinit (NOT by nfinit) flag is optional, and can be 0: default, compute zetak, or non-zero: compute the lambdak function, i.e. with the gamma factorszetakinitzetakinit(x): compute number field information necessary to use zetak, where x is an irreducible polynomialzncoppersmith(P, N, X, {B=N}): finds all integers x0 with |x0| <= X such that  gcd(N, P(x0)) > B. X should be smaller than exp((log B)^2 / (deg(P) log N)).znlog(x,g): g as output by znprimroot (modulo a prime). Return smallest non-negative n such that g^n = xznorderznorder(x,{o}): order of the integermod x in (Z/nZ)*. Optional o is assumed to be a multiple of the order.znprimroot(n): returns a primitive root of n when it existsznstar(n): 3-component vector v, giving the structure of (Z/nZ)^*. v[1] is the order (i.e. eulerphi(n)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generatorsSy>time for ct = %ld : %ld
time [max,t12,loop,reds,fin] = [%ld, %ld, %ld, %ld, %ld]
lllgramallgenincrementalGSgenlllintlllint_markedk = K%ld lllint[1], kmax = %ldlllint[2], kmax = %ld K%ldlllfp[1]dependent vectors in lllfplllfp (exact)count_max = %ld
lllfp giving upk =
...LLL reducing precision to %ld

Recomputing Gram-Schmidt, kmax = %ld
lllfp[1], kmax = %ld
Checking LLL basis...in precision %ld

Checking LLL basis
lllfp[2], kmax = %ldlllintpartialtm1 = %Zmid = %Zlllintpartialallnpass = %ld, red. last time = %ld, log_2(det) ~ %ld

negative bound in zncoppersmithzero polynomial forbiddenModified P: %Z
bound too largedelta = %d, t = %d, cond = %lf
Init: trying delta = %d, t = %d
Matrix to be reduced:
%Z
Entering LLL
bitsize bound: %ld
expected shvector bitsize: %ld
Candidate: %Z
bitsize Norm: %ld
Increasing dim, delta = %d t = %d
Roots: %Z
negative accuracy in lindep2qzer[%ld]=%ld
overflow in real shiftpslqInitialization time = %ld
pslqL2inconsistent primes in plindepnot a p-adic vector in plindepalgdep0negative polynomial degree in algdephigher degree than expected in algdepmaximal number of vectors must be providedbound = 0 in minim2not a definite form in minim0negative number of vectors in minim0adding vector = %Z
vector in new basis = %Z
base change matrix =
minim0, rank>=%lddimension 0 in fincke_pohstfirst LLL: prec = %ld
Fincke-Pohst, final LLL: prec = %ld
smallvectors looking for norm < %Z
smallvectorsNew bound: %Zsorting...
final sort & check...

$$$$mw~6bbRQ??I@yPD?-C6?ffffff?No such elliptic curveIncorrect curve name in ellconvertnameIncorrect vector in ellconvertname%s/elldata/ell%ldElliptic curves files not available for conductor %ld
[missing %s]Elliptic files %s not compatible
Incorrect curve name in ellsearchIncomplete curve name in ellsearchNo such elliptic curve in databasecan't deflateunexpected characterunknown function or error in formal parametersvariable name expectedobsolete functionerror opening invalid flagWarning:Warning: increasing precWarning: failed toaccuracy problemsbug insorry,sorry, not yet available on this systemcollecting garbage inprecision too lowincorrect typeinconsistent dataimpossible assignment I-->Simpossible assignment I-->Ithe PARI stack overflows !length (lg) overflowexponent (expo) overflowvaluation (valp) overflowoverflow in R->dbl conversionnon invertible matrix in gaussnot a square matrixunknown identifier valence, please reportnot an integer argument in an arithmetic functionnot enough precomputed primesnot enough precomputed primes, need primelimit ~ impossible inverse modulo: constant polynomialnot a polynomialzero polynomialtoo many iterations for desired precision in integration routinenot a definite matrix in lllgrambad argument for an elliptic curve related functiondivision by zerotrying to overwrite a universal objectnot enough memoryinfinite precisionnegative exponentnon quadratic residue in gsqrtwhat's going on ?can't allow allocatemem() in loopsinteger too bignot a suitable VECSMALL componentincorrect type or length in matrix assignmentunfinished stringbreak not allowed.%s = this function uses a killed variablelocal(); id too long in a stringified flaga stringified flag does not start with an idnumeric id in a stringified flagUnrecognized id '%s' in a stringified flagCannot negate id=value in a stringified flagUnrecognized action in a templateNon-numeric argument of an action in a templateJunk after an id in a stringified flagusing obsolete function %scan't kill thatexpected character: '%c' instead ofglobal variable not allowedskipidentifier (unknown code)too many parameters in user-defined function call; or ] expectedthis code has to come firstunknown parser code[install] identifier '%s' already in use[install] updating '%s' prototype; module not reloadednot a valid identifiercan't pop gp variablepanicno more variables availableglobal variable: %s already exists with incompatible valencerenaming a GP variable is forbiddenvariable number too bigincorrect type in %sbreak not allowed here (reading arguments)not a variable:not enough flags in string function signaturebreak not allowed in print()break not allowed here (expanding string)can't derive thisformal derivationbreak not allowed in O()break not allowed in test expressionsymbol already in usebreak not allowed here (defining global var)break not allowed here (reading function args)locallocal() bloc must appear before any other expressionunknown function '%s', expected '=' instead ofuser function %s: variable %Z declared twicebreak not allowed after !break not allowed after #truc(): n = %ldincorrect vector or matrixI can't remember before the big bangnot a proper member definitionbreak not allowed after ^this should be an integernot an integershift operand too bigseqcan't modify a pre-defined member: unknown member functionunused characters[+++]unused characters: %sbreak not allowed in assignmentbreak not allowed here (reading long)array index (%ld) out of allowed range [none]%s[1-%ld]a 0x0 matrix has no elementsbreak not allowed in array contextpositive integer expectedunknown functiononly functions can be aliasedcan't replace an existing symbol by an aliasa1a2a3a4a6areab2b4b6b8bidc4c6codiffcycfutuindexorderstatetufuzkstxbSbST~SUS
d'@Bʚ;l <= 2 in greffegtolongcomparisonp = 1 in Z_lvalremforbidden divisor %Z in ggvalforbidden or conflicting type in gvalnot an integer modulus in cvtopcvtop2gcvtopgexpogaffect (gen_0)gaffect (gen_1)gaffect (gen_m1)gaffect (gen_2)gaffect (gnil)gaffect (gpi)gaffect (geuler)gaffect (ghalf)gaffect (gi)gaffect (pol_1/pol_x)gtofpnormalizenormalizepolgsigneempty vector in vecmaxempty vector in vecminabs is not meromorphic at 0gabsnegative length in listcreatenegative index (%ld) in listputno more room in this L (size %ld)bad index in listinsertno more room in this listgtolistlistconcatpcpcycddc(d<dycpcpcdYddddbdbdbdtddddddddNeeeffeeeeeoookoppooAoCpCppkoppppppppppppprqqqq$r$rrprrrqqqrrfr}sst>u<vqt<vuts<v<vcu<vttuuu<vuu8vmvvwwvvv*wqwfx|x
y||@y|~yKz{|l|||||||<€	YȄ
/f+;++++++++̊#B;99999qڏ%5ˑoҎD:ďُ/ʑʑʑŖʑŖjt~"JYΘԘ[Hߚ`ߚߚ֚gg8q88Y888888different modulus in ff_poltypedifferent pointers in ff_poltypegmul2nnormalizing a series with 0 leading termincompatible variables in gredgmulsgtn}ĿHYcp]W::::E::rS"a!3kiwp7
${
x}hZvkAx 7X`cMSx?fff;	i+m5=FPu-r3"S s   !!3"!3"3"3"!!!v""""&##.$u$$&%&~%&&&Z&Z&Z&integer too largeI was expecting an integer heredefault: inexistent format%c%ld.%ld   format = %c%ld.%ld
1, 6, 3, 4, 5, 2, 3[1,,1], [5,,1], [3,,1], [7,,1], [6,,1], , [2,,1]1, 5, 3, 7, 6, 2, 3darkbglightbgboldfgexpected character: ']'[%ld,,%ld][%ld,%ld,%ld]   colors = "%s"
\ifx\%s\undefined
  \def\%s{%s}\fi
none   datadir = "%s"
   path = "%s"
tex2mail -TeX -noindent -ragged -by_parprettyprinter[secure mode]: can't modify '%s' default (to %s)yesbroken prettyprinter: '%s'   prettyprinter = "%s"
   prompt%s = "%s"
   help = "%s"
get_sep: argument too long (< %ld chars)default: incorrect value for %s [0:off / 1:on]   %s = 1 (on)
   %s = 0 (off)
[secure mode]: Do you want to modify the 'secure' flag? (^C if not)
secureechostrictmatchtimerfactor_add_primesnew_galois_formatarguments must be positive integersdefault: incorrect value for %s [%lu-%lu]realprecision   realprecision = %ld significant digits (%ld digits displayed)   %s = %lu %s
   %s = %lu
significant termscompatibleuser functions re-initialized(no backward compatibility)(warn when using obsolete functions)(use old functions, don't ignore case)(use old functions, ignore case)debugreadlinedebugfilesdebugmemhistsize   [logfile was "%s"]
logfile\hskip 0pt plus \hsize\relax\discretionary{}{}{}}PARIbreak\vskip\medskipamount\bgroup\bfPARIpromptSTART\egroup\bgroup\ttPARIpromptEND\egroupPARIinputEND\vskip\smallskipamount$\displaystyle{\tt\%#1} = #2$PARIout\ifx\%s\undefined
  \def\%s#1#2{%s}\fi
TeXstyle(raw)(prettymatrix)(prettyprint)(external prettyprint)parisize   %s = "%s"
psfilecomment> unknown default: %s"/usr/local/lib/pari/gphelp"GPHELP? (bits 0x2/0x4 control output of \left/\PARIbreak)(off)(on)(on with colors)(TeX output)(bits 0x2/0x4 control matched-insert/arg-complete)colorsdatadirpathpromptprompt_contqy	O?yPD3@exact type in subgrouplistsubgroup: index bound must be positive %ld     column selection:not a group in forsubgroupinfinite group in forsubgroup(lifted) subgp of prime to %Z part:

group:    lambda =     lambda'=     mu =     mu'=   alpha_lambda(mu,p) = %Z
  subgroup:  countsub = %ld
for this type  alpha = %Z
forsubgroup (alpha != countsub)nb subgroup = %ld
negative degree in legendreargument must be positive in polcyclonot a series in laplacenegative valuation in laplacenot a series in convoldifferent variables in convolprecision<=0 in gprectwo abcissas are equal in polintnot vectors in polinterpolatedifferent lengths in polinterpolateno data in polinterpolatenot a set in setsearchnot a set in setintersectnot a set in setminusdoubling stack in dirmul
not an invertible dirseries in dirdivinvalid bound in randomn too small (%ld) in numtopermnot a vector in permtonumreverse polmod does not existnot a polmod in modreversegen_sortincorrect lextype in vecsortnegative index in vecsortindex too large in vecsortnot a set in setunionegggrgegrgrgfgrggggrgrgrgconductor too large in ArtinNumber* Root Number: cond. no %ld/%ld (%ld chars)
diff(CHI) = %ZRecCoeff (cf = %ld, B = %Z)
RecCoeffRecCoeff3: no solution found!
Not enough precomputed primes (need all p <= %ld)* conductor no %ld/%ld (N = %ld)
	Init: 	character no: %ld (%ld/%ld)
S&TquickpolCompute WN0 = %ld
S & TN0 in QuickPol: %ld 
zetavalues = %Z
Checking the square-root of the Stark unit...
polrelnum = %Z
Compute %sIt's not a square...
Compute polrelnumstark (computation impossible)AllStarkpolrel = %Z
RecpolnumLooking for a modulus of norm: 
Trying modulus = %Z and subgroup = %Z
CplxModuluscpl = 2^%ld
Trying to find another modulus...No, we're done!
Modulus = %Z and subgroup = %Z
incorrect character in bnrrootnumbermain variable in bnrstark must not be xbase field not totally real in bnrstarkincorrect subgrp in bnrstarkclass field not totally real in bnrstarknew precision: %ld
Compute Cl(k)quadhilbertrealFindModulusthe ground field must be distinct from Qincorrect subgroup in bnrL1no non-trivial character in bnrL1@,@)\(4@ffffff?P@element_mulinot an integer exponent in nfpownegative power in element_powid_mod_pelement_mulidnot the same number field in basistoalgalgtobasis_inot the same number field in algtobasisincompatible variables in algtobasisnfmulelement_invelement_divnfdivelement_sqrelement_mulelement_pownot a matrix in matbasistoalgnot a matrix in matalgtobasisnot the same number field in rnfalgtobasisarch_to_permzero element in zsigneelement_invmodidealmodule too large in Fp_shanksFp_shanks, k = %lda not invertible in ff_PHlog_Fpnf_Pohlig-Hellman: DL mod %Z^%ld
module too large in ffshanksentering zlog, with a = %Z
leaving
not an element of (Z/pZ)* in znlogincorrect archimedean component in IdealstarIdealstar needs an integral non-zero ideal: %Ztreating pr^%ld, pr = %Z
  treating a = %ld, b = %ld
zidealij done
%Z not a nfeltnot an element in zideallogIdeallist9,,,9,,,,,,,,,,,,,,(,4+=iii=iiiiiOiOiiiiiiiii_inot an n-th power in idealsqrtnreducing beta = %Z
beta reduced via ell-th root = %Z
reducebetabeta LLL-reduced mod U^l = %Z
naive reduction mod U^l: unit exp. = %Z
beta reduced = %Z
main variable in kummer must not be x[rnfkummer] conductorkummer for composite relative degreebug%d in kummerStep 1
polred(compositum) = %Z
Step 2
[rnfkummer] compositumStep 3
[rnfkummer] bnfinit(Kz)[rnfkummer] Selmer groupStep 4
Step 5
isvirtualunitStep 6
Step 8
Step 9, 10 and 11
Step 12
Step 13
Step 14, 15 and 17
Step 16
Step 18
[rnfkummer] candidate listComputing Newton sums: %ld(%ld) polrel(beta) = %Z
1, 0, 
[%s/%s %lu %luMPQS: wrapping, more primes for A now chosen near FB[%ld] = %ld
MPQS: new bit pattern for primes for A: 0x%lX
MPQS: chose primes for A FB[%ld]=%ld%serror whilst writing to file %sMQPS: short of space -- another buffer for sorting
MQPS: line wrap -- another buffer for sorting
MPQS: relations file truncated?!
MPQS: done sorting one file.
error whilst flushing file %scannot rename file %s to %sMPQS: renamed file %s to %s
 : 1 1 %ld %ldMPQS: combining
    {%ld @ %s : %s}
  * {%ld @ %s : %s}
 == {%s}
MPQS: combined %ld full relation%s
cube7th power5th powerMPQS: decomposed a square
MPQS: decomposed a %s
longershorter, looking for more...comp.unknownftell error on full relations fileMPQS: full relations file %s than expectedMPQS panicking\\ MATRIX READ BY MPQS
FREL=\\ KERNEL COMPUTED BY MPQS
KERNEL=MPQS: Gauss done: kernel has rank %ld, taking gcds...
MPQS: no solutions found from linear system solvercannot seek FREL fileFREL file truncated?![1]: mpqs_solve_linear_systemMPQS (relation is a nonsquare)[2]: mpqs_solve_linear_systemMPQS: X^2 - Y^2 != 0 mod N
	index i = %ld
MPQS: wrong relation found after GaussMPQS: splitting N after %ld kernel vector%s
MPQS: got two factors, looking for more...
MPQS: resplitting a factor after %ld kernel vectors
MPQS: got %ld factors%s
[3]: mpqs_solve_linear_systemMPQS: wrapping up vector of %ld factors
	packaging %ld: %Z ^%ld (%s)
manyseveral and combiningMPQS: number to factor N = %Z
MPQS: number too big to be factored with MPQS,
	giving upMPQS: factoring number of %ld decimal digits
MPQS: found multiplier %ld for N
MPQS: factoring this number will take %s hours:
N = %ZMPQS: kN = %Z
MPQS: kN has %ld decimal digits
MPQS: Gauss elimination will require more than
	128MBy of memory	(estimated memory needed: %4.1fMBy)
MPQS: creating factor base and allocating arrays...
MPQS: precomputing auxiliary primes up to %ld
MPQS: FB [-1,2,<%lu>,%lu...] Wait a second --
,%lu
MPQS: found factor = %ld whilst creating factor base
MPQS: sizing out of tune, FB size or tolerance
	too largeMPQS: computing logarithm approximations for p_i in FB
MPQS: sizing out of tune, FB too small or
	way too few primes in AMPQS: sieving interval = [%ld, %ld]
MPQS: size of factor base = %ld
MPQS: striving for %ld relations
MPQS: coefficients A will be built from %ld primes each
MPQS: primes for A to be chosen near FB[%ld] = %ld
MPQS: smallest prime used for sieving FB[%ld] = %ld
MPQS: largest prime in FB = %ld
MPQS: bound for `large primes' = %ld
MPQS: sieve threshold = %u
MPQS: first sorting at %ld%%, then every %3.1f%% / %3.1f%%
MPQS: starting main loop
MPQSLPTMPFRELFNEWLPRELLPNEWCOMBMPQS: chose Q_%ld(x) = %Z x^2 %c %Z x + C
MPQS: Ran out of primes for A, giving up.
MPQS: found %lu candidate%s
%s @ %s :%s

MPQS: passing the %3.1f%% sort point, time = %ld ms

MPQS: passing the %3.1f%% sort point

MPQS: split N whilst combining, time = %ld ms
MPQS: found factor = %Z
MPQS: done sorting%s, time = %ld ms
MPQS: found %3.1f%% of the required relations
MPQS: found %ld full relations
MPQS:   (%ld of these from partial relations)
MPQS: Net yield: %4.3g full relations per 100 candidates
MPQS:            %4.3g full relations per 100 polynomials
MPQS: %4.1f%% of the polynomials yielded no candidates
MPQS: next sort point at %3.1f%%

MPQS: starting Gauss over F_2 on %ld relations

MPQS: time in Gauss and gcds = %ld ms
MPQS: found factors = %Z
	and %Z
MPQS: found %ld factors =
	%Z%s
MPQS: no factors found.

MPQS: restarting sieving ...

MPQS: giving up.
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'
0
9
B
K
+_ee+rm&`!` m&m&""##$o$$^$5D:))/)N*):)~+^,/x-/--...//u.111$282521C348o2811666Q6g77embedded braces (in parser)unexpected closing braceTeX variable name too long	-#<%d>read failedwrite failedthis object uses debugging variablesout of range in integer -> character conversion (%ld)%s is not a GP binary file%s not written for a %ld bit architectureunexpected endianness in %s%s written by an incompatible version of GP---- (type RETURN to continue) ----%c[0m%c[%ld;%ldm%c[%ld;%ld;%ldmCOLUMNSLINESt_INTt_REALt_INTMODt_FRACt_COMPLEXt_PADICt_QUADt_POLMODt_POLt_SERt_RFRACt_COLt_MATt_LISTt_STRt_VECSMALLunknown type %ld[;]
]

%020ld,CLONEint = pol = NULL
gen_0
[&=%016lx] %s(lg=%ld%s):%016lx  chars:(%c,lgefint=%ld):(%c,expo=%ld):(precp=%ld,valp=%ld):(%c,varn=%ld):(%c,varn=%ld,prec=%ld,valp=%ld):(lgeflist=%ld):mod = num = den = real = imag =   p : p^l :   I : coef of degree %ld = %ld%s component = %ld%s column = mat(%ld,%ld) = :
   hash = %3ld, valence = %3ld, menu = %2ld, code = %s
   next = %s no such functioninvalid range in print_functions_hash*** hashcode = %lu
%3ld:%3ld 
 Top : %lx   Bottom : %lx   Current stack : %lx
 Used :                         %ld  long words  (%ld K)
 Available :                    %ld  long words  (%ld K)
 Occupation of the PARI stack : %6.2f percent
 %ld objects on heap occupy %ld long words

 %ld variable names used out of %d

0.E%ldVecsmall([Mod(mod(qfr(qfi(matrix(0,%ld)matrix(0,%ld,j,k,0)Mat(mat( + O(Qfb(List([[;] - List( mod  / / []
^{%ld}\*\frac{}{\cdot\pmatrix{ \cr}
\cr
\pmatrix{
 \cr
  ( \left(\right) \PARIbreak  +closedeleteclose pipeI/O: closing file %s (code %d) 
I/O: checking output pipe...
                  
I/O: new pariFILE %s (code %d) 
could not open requested file %sI/O: opening file %s (mode %s)
tempfile %s already existsI/O: can't remove file %sI/O: removed file %s
run-away string. Closing itrun-away comment. Closing itgp_readvec_stream: reaching %ld entries
gp_readvec_stream: found %ld entries
I/O: leaked file descriptor (%d): %s[pipe:] '%s' failedskipping directory %s%s.gpcan't expand ~unknown user %s%sundefined environment variable: %s[secure mode]: about to write to '%s'. OK ? (^C if not)
.:~:~/gpYou never gave me anything to read!input%s is a GP binary file. Please use writebinmalformed binary file (no name)setting %s
unknown code in readobjbinary output%ld unnamed objects read. Returning then in a vectorI can't see into the future%s is set (%s), but is not writeable%s is set (%s), but is not a directory/var/tmpGPTMPDIR.%ld.%ld%.8s%scouldn't find a suitable name for a tempfile (%s)couldn't find a suitable name for a tempdir (%s),b7@valuation overflow in sqrtnmpsc1incorrect precision in transca transcendental functiondegree overflow in pow_monomesqrtrodd exponent in p-adic sqrtpadic_sqrtnon positive argument in mplogsqrtnrnot a p-adic argument in teichmullerzero argument in palogcan't compute tan(Pi/2 + kPi)non zero exponent in gsincosp-adic argument out of range in gexpgpow: need integer exponent if series valuation != 0gpow: 0 to a forbidden powergpow: 0 to a non positive exponentgpow: underflow or overflowgpow: modulus %Z is not primegpow: nth-root does not existser_pow%Z should divide valuation (= %ld) in sqrtnzero argument in mploggloglog is not meromorphic at 0agm of two vector/matricessecond arg must be integer in gsqrtn1/0 exponent in gsqrtnnth-root does not exist in gsqrtnp-adic argument out of range in gcosp-adic argument out of range in gsingtangcotan0 argument in cotanqTqTTqTUqTqTqTTTTUTUUUUUUUUUaVcVcbVcVcVcXbbebVcVcebVcVcVcbbb@eiefiifiihiiGh>lljlll4kllllllll&1MiD,,C,G,&4&>9ݽ¾l̾4m*w9B.F@@+eG?9B.?8,6V?UUUUUU%@9B.&@|?5^@Choosing t = %ld
aprcl: e(t) too smallJacobi sums and tables computed
Step4: q-values (# = %ld, largest = %ld): 
Step5: testing conditions lp
aprcl test fails! this is highly improbableStep6: testing potential divisors
Individual Fermat powerings:
  %-3ld: %3ld
Number of Fermat powerings = %lu
Maximal number of nondeterministic steps = %lu
wrong modulus in galoissubcyclogenerators must be prime to conductor in galoissubcyclowrong type in galoissubcycloSubCyclo: testing %ld^%ld
SubCyclo: %ld not found
SubCyclo: new conductor:%ld
SubCyclo: conductor:%ld
Subcyclo: prime l=%Z
Subcyclo: borne=%Z
Subcyclo: val=%ld
padicsqrtnlift.bnr must be over Q in bnr_to_znstardegree <= 0 in galoissubcycloPlease do not try to break PARI with ridiculous counterfeit data. Thanks!not a HNF matrix in galoissubcycloN must be a bnrinit or a znstar if H is a matrix in galoissubcycloMatrix of wrong dimensions in galoissubcycloSubcyclo: elements:Subcyclo: complex=%ld
znstar_conductorznstar_cosetsSubcyclo: orbits=%Z
Subcyclo: %ld orbits with %ld elements each
degree does not divide phi(n) in subcyclonon-cyclic case in polsubcyclo: use galoissubcyclo insteadsubcyclo_rootssubcyclo_cyclicroots_to_polzellw1 and w2 R-linearly dependent in elliptic functionreduction mod SL2 (reduce_z)weipellnumnot a prime in localredellinit data not accurate enough. Increase precision%lu is not prime, use ellakapell (f^(i*s) = 1)localred (p | c6)localred (nu_D - nu_j != 0,6)singular curve in ellinitnot a rational curve in ellintegralmodel%Z - (%Z)
incompatible p-adic numbers in initellvaluation of j must be negative in p-adic ellinitinitell for 2-adic numbersk not a positive even integer in elleisnumcan't evaluate ellzeta at a polecan't evaluate log(ellsigma) at lattice pointbadgood  z  = %Z
  z1 = %Z
  z2 = %Z
ellpointtoz: %s square root
CM_ellpowpowell for nonintegral CM exponentnorm too large in CMnot a complex multiplication in powellpowell for non integral, non CM, exponentsexpecting a simple variable in ellwpnot an integral curve in elllocalredprime too large in apell2, use apell[apell1] baby steps, s = %ld[apell1] sorting[apell1] giant steps, i = %ldnot a prime in apellnot an integral modelanell for n >= %lunot an integral model in akellcut-off point must be positive in lseriesellpoint not on elliptic curvetwo vector/matrix types in bilhellorderell for nonrational elliptic curvestorselltorsell (bug1)torsell (bug2)torsell (bug3)PQRPPQPQQQQ"TTTSSSS;T+S7SdSS/YYXXXKYWW;WKYQYVVVVVVVZZ~ZZZZZZLZ@G?zG!"@@X[#!"@sqred2gsmithallhess, m = %ldgnorml1QuickNormL1gconjgnormgnorml2incompatible field degrees in conjvecnot a rational polynomial in conjvecassmatgtraceeasycharincorrect variable in caradjnot a positive definite matrix in sqred1sqred3more rows than columns in matrixqzmatrix of non-maximal rank in matrixqzmatrixqz when the first 2 dets are zeronot a rational matrix in matrixqzincompatible matrices in hnf_specialhnf_special[1]. i=%ldhnf_special[2]. i=%ldPermutation: %Z
matgen = %Z
non coprime ideals in hnfmergehnf[1]. i=%ldhnf[2]. i=%ldallhnfmodnb lines > nb columns in hnfmodallhnfmod[1]. i=%ldallhnfmod[2]. i=%ldhnflllhnflll (reducing), i = %ldhnflll, k = %ld / %ldEntering hnffinal:
dep = %Z
mit = %Z
    hnflll done
hnffinal, i = %ld    first pass in hnffinal done
Leaving hnffinal
mit = %Z
B = %Z
C = %Z
    1st phase done
    2nd phase done
hnfadd (%ld + %ld)H = %Z
C = %Z
Entering hnfspec
    after phase1:
hnfspec[1]hnfspec[2]    after phase2:
hnfspec[3], (i,j) = %ld,%ld    matb cleaned up (using Id block)
hnfspec [%ld x %ld] --> [%ld x %ld]mathnfspec with large entriesextendedgcdhnfpermhnfallhnfall[1], li = %ldhnfall[2], li = %ld
hnfall, final phase: hnfall[3], j = %ldmatrixqz_auxmatrixqz2matrixqz3matrixqz0non integral matrix in smithallstarting SNF loop
i = %ld: [1]: smithall i = %ld[2]: smithall, i = %ld[3]: smithallsmithcleanaccuracy lost in matfrobeniusvariable must have higher priority in matfrobenius~~~~~'''ggggYgK3GGG
[

[0[0[[[[[[[__			m((						P					--
	-

---------
333333?greal/gimaggvargpolvarnot the same prime in padicpreciscomplex%gdivmodnon-positive argument in O()zero argument in O()incorrect object in O()incorrect argument in O()gfloor\gceilgroundgrndtoigfloor2nisintgtrunca log appears in intformala log/atan appears in intformalintegvariable must have higher priority in gtopolyt_SER with negative valuation in gtopolymain variable must have higher priority in gtoservectosmallthis object is a leaf. It has no componentsnonexistent componentnon existent component in truecoeffnonexistent component in truecoeffnumerlift_internevaluation of a power seriesgevalsimplify_iincorrect permtuation to inverseinv_serinvalid data in qfevalinvalid quadratic form in qfevalinvalid vector in qfevalinvalid data in qfbevalinvalid quadratic form in qfbevalinvalid vector in qfbevalinvalid data in qf_base_changeinvalid base change matrix in qf_base_changeinvalid data in hqfevalinvalid quadratic form in hqfevalinvalid vector in hqfevalpolevalpoleval: i = %ldforbidden substitution by a non square matrixforbidden substitution by a vectorforbidden substitution in a scalar typegsubstnon positive valuation in a series substitutionnon polynomial or series type substituted in a seriessubst: unexpected variable precedencedifferent number of variables and values in substvecnot a variable in substvecnot a series in serreversevaluation not equal to 1 in serreverse֣G֣֣֣99-9-#-(@@U@UKUPeepepapjӨӨd//nNwWWWWWWWWWܯNNNϯܯϯϯܯNϯNNN%0:@FY:v@̰,,,f{{`U8ER8iIS]I

^
0GGGG		R]				Vik4xff'HHAXu440)+))))pPL:::V::::EEEE.8Ehb..EEEEE/*D2,P] ] ] ]    GDefault bound for regulator: 0.2
  *** testing p = %lu
     p divides h(K)
     p divides w(K)
     Beta list = %Z
       generator of (Zk/Q)^*: %Z
       prime ideal Q: %Z
       column #%ld of the matrix log(b_j/Q): %Z
       new rank: %ld
incorrect subgroup in %swrong type in too_bigLbnrclassnobnrdiscrayBuchrayneither bnf nor bnr in conductor or discraybad subgroup in conductor or discraysmith/class groupplease apply bnrinit(,,1) and not bnrinit(,,0)isprincipalrayMinkowski bound is too largelarge Minkowski bound: certification will be VERY longSearching minimum of T2-form on units:
   BOUND = %ld
M* = %Z
old method: y = %Z, M0 = %Z
[ %ld, %ld, %ld ]: %Z
(lower bound for regulator) M = %Z
Mahler bound for regulator: %Z
sorry, too many primes to check
PHASE 2: are all primes good ?

  Testing primes <= B (= %lu)

  Testing primes | h(K)

not an Abelian extension in rnfnormgroup?non Galois extension in rnfnormgroupnot an Abelian extension in rnfnormgroupincorrect character length in KerCharfactordivexact is not exact!discrayabslistnot a factorisation in decodemoduleincorrect hash code in decodemodulenon-positive bound in Discrayabslistr1>15 in discrayabslistarchStarting zidealstarunits computations

Starting bnrclassno computations
[1]: discrayabslistarchStarting discrayabs computations
[2]: discrayabslistarch999:
::N:U:vq
-?̯?
W8z?-@x?QsX@N9@4s@dmlv@z	@CX~
@R%r@&c@)Im@3x@8k	\7@Hء@T@@5S@ߦ޿!@@4mI@%PH[v@Sbb@&	A $@z"@!Q!@PE\n @V*@>L;'@A
B%@
g$@*}2#@cq"@
S!@e}b*@u[`)@W-'@
ZZ&@ȥ;q%@=ˎF$@
ԕ-@D7H,@mQ+@b)@/(@MNL'@o
l`&@RFi0@DH~/@)&o/.@A"pj,@㐠+@Iqa*@Fw()@˱a2@>_1@Eկt>0@BF0@r.@ă-@(B,@$("+@Mt3@k3@h#M)U2@}1@&00@GX0@eh/@Zh9#.@Rk\5@
^4@v)3@)K3@W2@p1@U~L1@O}50@Xu`0@U>	7@`]9U6@Ċ5@vJ14@DZB4@?3@/2@9qޤ@2@n˙1@o8@-P8@,AN7@8	6@
O5@t9f95@ȿ+V4@#G:3@@153@C^2@"k:@+j0
ó9@ЗE8@bG8@RK)7@xT6@/%06@tHb5@8:4@x *4@;_ <@qVqf;@<\iD:@c9@H*@9@֫而8@RI97@HPW(7@0(x6@uIh&5@Vl5@?rZ|
?ư>{Gz?insufficient precision for p = 2 in hilbertlist of numerators too short in sfcontf2sfcont2negative integer in sqrtintexponent overflow in regulanegative nmax in sfcontintegral part not significant in sfcontfundunitassociationisanypowerisanypower: now k=%ld, x=%Z
missing exponentZ_issquareclassno2discriminant too large in classnodiscriminant too big in classnoclassno with too small orderp = 1 in hilbert()forbidden or incompatible types in hilcomposite modulus in Fl_sqrt: %lularge exponent in Mod(a,N)^n: reduce n mod phi(N)not an element of (Z/nZ)* in orderprimitive root mod 2^%ld does not existzero modulus in znprimrootprimitive root mod %Z does not existispower for non-rational argumentsnot a prime in Fp_sqrtcomposite modulus in Fp_sqrt: %Z1/0 exponent in Fp_sqrtnFp_sqrtlgisprimenegative argument in factorial functioncontfrac0incorrect size in pnqnbestappr0incorrect bound type in bestappr33333O>QYzoo6I	 	 	 	 	 	 	 	 	 E,W@Gz?truncr (precision loss in truncation)floorr (precision loss in truncation)NaN or Infinity in dbltorrtodblratlift: bmax must be > 0, found
	bmax=%Z
ratilft: amax must be >= 0, found
	amax=%Z
ratlift: must have 2*amax*bmax < m, found
	amax=%Z
	bmax=%Z
	m=%Z
ratlift: must have 0 <= x < m, found
	x=%Z
	m=%Z
ratlift failed to catch d1 == 0
@

	prime_loop_initforparistep equal to zero in forstepnot a vector in forvecnot a vector of two-component vectors in forvecnon integral index in sumnon integral index in suminfnon integral index in prodinfnon integral index in prodinf1constant term != 1 in direulernegative number of components in vectoridentical index variables in matrixnegative number of columns in matrixnegative number of rows in matrixnon integral index in sumaltnon integral index in sumposnon integral index in sumpos2roots must be bracketed in solvetoo many iterations in solve@@?ףp=
?sorry, too many block systems in nfsubfieldsp = %ld,	lcm = %ld,	orbits: %Z
Chosen prime: p = %ld
relatively prime polynomials expectedTR_POL(1), i = %ld/%ldTR_POL(-1), i = %ld/%ldTR_POL, i = %ld/%ldEntering compute_data()

f = %Z
p = %Z, lift to p^%ld
2 * Hadamard bound * ind = %Z
2 * M = %Z
delta[%ld] = %Z
d-1 test failed
pol. found = %Z
coeff too big for pol g(x)
changing f(x): p divides disc(g)
candidate = %Z
lifting embedding mod p^k = %Z^%ld
coeff too big for embedding
embedding = %Z
lg(Z) = %ld, lg(Y) = %ld
Z = %Z
Y = %Z

ns = %ld
overflow in calc_block
* Look for subfields of degree %ld


Subfields of degree %ld: %Z

***** Entering subfields

pol = %Z

***** Leaving subfields


Time %s rel [#rel/#test = %ld/%ld]: %ld
[quadhilbert] incorrect values in pq: %luqrf5_rho_powpowsubFBquadclass number = %ldquadhilbert (pq)quadhilbertimag (can't find p,q)p = %lu, q = %lu, e = %ld
product, error bits = %ldquadhilbertimagincorrect data in findquadnot a polynomial of degree 2 in quadhilbertquadhilbert needs a fundamental discriminantform_to_idealnot a polynomial of degree 2 in quadrayquadray needs a fundamental discriminantquadray: looking for [a,b] != unit mod 2f
[a,b] = [%ld,%ld] lambda = %Z
get_lambdacomputeP2sorry, couldn't deal with this field. PLEASE REPORT
*** Bach constant: %f

rel = %ld^%ld cglob = %ld. Bach constant <= 0 in buchquadfactor baseFB = %Z
subFBquad (%ld elt.)*** Changing sub factor base
KC = %ld, need %ld relations
...need %ld more relations
initialreal_relations %ldP
#### Tentative class number: %Z
regulator is zero.
#### Tentative regulator: %Z
be honest for primes from %ld to %ld
be honestincorrect parameters in quadclassunit@&DT!	@LXz????subresallMignotte bound: %Z
Beauzamy bound: %Z
polsym of a negative npolsym_genLLL_cmbf [no factor]S_2   bound: %Z^%ld
coeff bound: %Z^%ld
need positive degree in gdeflatecan't deflate this power series (d = %ld): %Zleftright_powprod: remaining objects %ld
not a factorisation in factorbackmissing nf in factorbackeltRgX_gcd_simpleQ_denomQ_muli_to_intQ_divmuli_to_intQ_div_to_intmissing case in gdivexactMultiLift: bad argsbuilding treelifting to prec %ldnot a polynomial in polhenselliftnot a factorization in polhenselliftnot a prime number in polhenselliftnot a positive exponent in polhenselliftnot an integral polynomial in polhenselliftnot an integral factorization in polhenselliftnot a correct factorization in polhenselliftpolhensellift: factors %Z and %Z are not coprimepseudorem dx = %ld >= %ldpseudodiv dx = %ld >= %ldnot the same variables in sylvestermatrixsrgcdsrgcd: dr = %ld
subresall, dr = %ldsubresext, dr = %ldinexact computation in subresextresultantducos, degpol Q = %ldnextSousResultant j = %ld/%ldRgX_extgcd, dr = %ldinexact computation in RgX_extgcdQ_contentHensel lift (mod %Z^%ld)
### K = %d, %Z combinations
|
found factor %Z
remaining modular factor(s): %ld
Naive recombinationlast factor still to be checked

LLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)
LLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld
for this block of tracesLLL_cmbf: rank decrease
special_pivot output:
%Z
LLL_cmbf: checking factor %ld
LLL_cmbf: chk_factors failed* Time LLL: %ld
* Time Check Factor: %ld
KnapsackRoot boundDDF_roots, m = %ldRecombination...tried prime %3ld (%-3ld %s). Time = %ld
DDF: wrong numbers of factorssplitting mod p = %ldTime setup: %ld
Total Time: %ld
===========
factpolnfrootsQdiscsrfactor for general polynomialsfactor of general polynomialcan't factor %Zpartial factorization is not meaningful heregisirreduciblereduceddiscsmithnon-monic polynomial in poldiscreducednot a squarefree polynomial in sturmpolsturm, dr = %ldnon-invertible polynomial in RgXQ_invginvmodmatratliftpolratliftnfgcd: p=%d
nfgcdAGGTGGGGǩGGGGGGǫǫǫǫǫǫCǫǫǫǫǫǫPyyyyyyyyyyyy444} Bl8]~^^}}||IFYFkFFFD||T|T||K||||||fffn$ % $'T !%"#    \'\'!"H"""@'@'"](44j(4?.444(44-((z)(++(d+ h|5@h㈵>@@fundamental units too largeinsufficient precision for fundamental unitsunknown problem with fundamental unitsbnfinit: %s%s, not given
m = %Z
Computing powers for subFB: %Z
powFBgencodeprimebase change =
# ideals tried = %ld
SPLIT: increasing factor base [%ld]
not a vector/matrix in cleanarch
#### Computing class group generators
classgroup generators
#### Computing fundamental units
getfuincorrect big number fieldred_mod_unitsnot a factorization matrix in isunitnot an algebraic number in isunitbnfnewpreczero ideal in isprincipalinsufficient precision for generators, not givenisprincipal (incompatible bnf generators)precision too low for generators, e = %ldprecision too low for generators, not givencompleting bnf (building cycgen)PHASE 1: check primes to Zimmert bound = %lu

**** Testing Different = %Z
     is %Z
*** p = %lu
  Testing P = %Z
    Norm(P) > Zimmert bound
    #%ld in factor base
End of PHASE 1.

compute_Rinitalg & rootsof1weighted G matricesBach constant <= 0 in buchR1 = %ld, R2 = %ld
D = %Z
LIMC = %ld, LIMC2 = %ld
########## FACTORBASE ##########

KC2=%ld, KC=%ld, KCZ=%ld, KCZ2=%ld
++ LV[%ld] = %Zsub factorbase (%ld elements)KCZ = %ld, KC = %ld, n = %ld

#### Looking for %ld relations (small norms)

*** Ideal no %ld: [%Z, %Z, %Z, %Z]
small_norm (precision too low)v[%ld]=%.4g BOUND = %.4g
for this idealsmall norm relations  small norms gave %ld relations.
  nb. fact./nb. small norm = %ld/%ld = %.3f

#### Looking for random relations

(more relations needed: %ld)
looking hard for %Z
relation cancelled: (jid=%ld,jdir=%ld)
++++ cglob = %ld: new relation (need %ld)for this relationbuchall (%s)
#### Computing regulator multiple

#### Computing check
truncation error in bestappr
D = %Z
den = %Z
bestappr/regulator
#### Tentative regulator : %Z

 ***** check = %f
Be honest for %ld primes from %ld to %ld
be_honest() failure on prime %Z
non-monic polynomial. Change of variables discardedincorrect parameters in classgroupclassgroupallcompleting bnf (building matal)*%ld makematal(@&DT!@+m0_? @4bi???333333?argument must belong to upper half-planenon-positive valuation in etabad argument for modular functionjbesseljbessel around a!=0p-adic jbessel functionkbesselkbessel around a!=0not an integer index in jbesselhp-adic jbesselh functionhyperu's third argument must be positivezero argument in a k/n bessel functionp-adic kbessel functioncannot give a power series result in k/n bessel functionincgam2non-real argument in eint1Entering veceint1:
negative or zero constant in veceint1nstop = %ld
inv_szeta_euler, p = %lu/%lutoo large negative arg %ld in zetaszetalim, nn: [%ld, %ld]
tab[q^-s] from 1 to N-1sum from 1 to N-1czetaBernoulli sumargument equal to one in zetagzetazeta of power seriestwistpartialzeta (1), j = %ld/%ldtwistpartialzeta (2), j = %ld/%ldtwistpartialzeta (3), j = %ld/%ldnegative index in polylogpadic polylogarithmpolylog around a!=0gpolyloggpolylogztrueetaq >= 1 in thetak < 0 in thetanullk))`)`)M)```````hhUhhhhhhhooooo}000333T5T533T5T5T5A5HH'IH:IH'IHH:I:I:I:I:I:I:IdJdJdJB
?Ǝ%N@9B.6@ۦx\T?p=
ף@e2@Jd?8dg?(\?-"l?UUUUUU?&DT!?Gz?Q?conformal_polrefine_Frefine_HFFTinitFFTcauchy_boundall_roots: restarting, i = %ld, e = %ld
isrealapprroots (conjugates)invalid coefficients in rootsjQ	@j@??@@p^_??ht@O贁N[?IG]??55?f0'u"@O贁Nk?i\??]9?@?&?r@0?	?{Gz?+eG@invalid polynomial in thue (need n>2)get_emb  errdelta = %Z
sol = %Z
Partial = %Z
  B0  = %Z
  Baker = %Z
non-monic polynomial in thueinithuec1 = %Z
c2 = %Z
Indice <= %Z
epsilon_3 -> %Z
invalid polynomial in thue (need deg>2)prec = %d
expected an integer in bnfisintnormgcd f_P  does not divide n_p
looking for a fundamental unit of norm -1
%Z eliminated because of sign
Non trivial conditional class group.
  *** May miss solutions of the norm equationx1 -> %Z
x2 -> %Z
c14 = %Z
* real root no %ld/%ld
  c10 = %Z
  c13 = %Z
  - norm sol. no %ld/%ld
  c6  = %Z
  c8  = %Z
  c11 = %Z
  c15 = %Z
  Entering CF...
    B0 -> %Z
CF failed. Increasing kappa
Semirat. reduction: B0 -> %Z
thue (totally rational case)  Entering LLL...
C (bitsize) : %d
LLL_First_Pass successful !!
LLL failed. Increasing kappa
Semirat. reduction: B0 -> %Z x <= %Z
Checking (\pm %Z, \pm %Z)
Not enough precision in thuenot a tnf in thueAll solutions are <= %Z
* Checking for small solutions
SmallSolsII
@RQ@@=
ףp=?Gz?H}8?Ů,?vecsmall_copyperm_to_GAPempty group in group_domaincoset not found in cosets_perm_searchgaloissubgroup: not a WSS groupwrong argument in galoisisabelianGroup(())Group(PermutationGroup<1|>PermutationGroup<Rg_to_FlRg_to_FpFFInit: using subcyclo(%ld, %ld)
powers is only [] or [1] in FpX_FpXQV_compoFpX_FpXQV_compo: %d FpXQ_mul [%d]
pol[Frobenius]matrix cycloZZ_%Z[%Z]/(%Z) is not a field in FpX_ffintersectnon invertible polynomial in FpXQ_invRg_to_FpXQFpXQ_sqrtl1/0 exponent in FpXQ_sqrtnFF l-Gen:next %Z
fflgennon-invertible polynomial in FpXQX_gcdbad degrees in FpX_ffintersect: %d,%d,%dFpM_kerPolynomials not irreducible in FpX_ffintersectpows [P,Q]FpM_invimageFpXQ_matrix_powsfactor_irred_matfactor_irredFpX_resultant (da = %ld)polint_triv2 (i = %ld)FpV_polintbound for resultant: 2^%ld
resultant mod %ld (bound 2^%ld, stable = %d)ZX_resultantdifferent variables in modulargcdgcd mod %lu (bound 2^%ld)bound 2^%ld. Goal 2^%ldmodulargcd: trial division failedZY_ZXY_resultant_all: LERS needs lambdaTrying lambda = %ld
bound for resultant coeffs: 2^%ld
Degree list for ERS (trials: %ld) = %Z
Final lambda = %ld
resultant mod %ld (bound 2^%ld, stable=%ld)ZY_ZXY_rnfequationQXQ_invQXQ_inv: mod %ld (bound 2^%ld)QXQ_inv: char 0 check failednon positive degree in ffinitX@\(\?Flx_to_Flvnon invertible polynomial in Flxq_invFlxqX_safegcdreducible form in qfr_rhozero discriminant in %ssquare discriminant in %sdiscriminant not congruent to 0,1 mod 4 in %snegative discriminant in %sqfr_unit_by_discpositive discriminant in %sqfi_unit_by_discreducible form in qfr_initnegative definite t_QFIShanks distance must be a t_REAL in qfrzero discriminant in Qfbdifferent discriminants in qfb_compqfr_unitnot a t_REAL in 4th component of a t_QFRqfi_unitnot an integer exponent in nupowredimagsl2redimagcompositionnot a t_QFI in powimagnot a t_QFI in nuduplnot a t_QFI in nucompnot a real quadratic form in redrealnot a t_QFR in powrealrawdiscriminant not congruent to 0,1 mod 4 in primeformcornacchiad must be 0 or 3 mod 4GaloisConj: Solution too large, discard it.
f=%Z
 borne=%Z
 l-borne=%Z
MonomorphismLift: trying early solution %Z
MonomorphismLift: true early solution.
MonomorphismLift: false early solution.
MonomorphismLift: lift to prec %dmonomorphismlift()Polynomial not squarefree in galoisinitGaloisAnalysis:non Galois for p=%ld
GaloisAnalysis:Nbtest=%ld,p=%ld,o=%ld,n_o=%d,best p=%ld,ord=%ld,k=%ld
Galois group almost certainly not weakly super solvableGaloisAnalysis:p=%ld l=%ld group=%ld deg=%ld ord=%ld
galoisanalysis()%d.frobenius powers4test()incorrect permutation in permtopolGaloisConj:Entree Init Test
GaloisConj:Sortie Init Test
A4GaloisConj:I will test %ld permutations
A4GaloisConj: %ld hop sur %ld iterations
A4GaloisConj:sigma=%Z 
A4GaloisConj:tau=%Z 
A4GaloisConj:orb=%Z 
A4GaloisConj:O=%Z 
A4GaloisConj:%ld hop sur %d iterations max
S4GaloisConj:Computing isomorphisms %d:%Z
S4GaloisConj:Testing %d/3:%d/4:%d/4:%d/4:%Z
S4GaloisConj:sigma=%Z
S4GaloisConj:pj=%Z
S4GaloisConj:Testing %d/3:%d/2:%d/2:%d/4:%Z:%Z
S4GaloisConj:Testing %d/8 %d:%d:%d
galoisconj2polconjugate %ld: %Z
incorrect denominator in initgaloisborne: %ZvandermondeinverseGaloisConj:val1=%ld val2=%ld
GaloisConj: Bound %Z
FixedField: Size: %ldx%ld
FixedField: Weight: %Z
FixedField: Sym: %Z
p too small in fixedfieldsympolFixedField: Found: %Z
GaloisConj:denominator:%Z
GaloisConj:Testing A4 first
GaloisConj:Testing S4 first
GaloisConj:p=%ld deg=%ld fp=%ld
Trying degre %d.
Galoisconj:Subgroups list:%Z
GaloisConj:I will try %Z permutations
Combinatorics too hard : would need %Z tests!
I will skip it, but it may induce an infinite loopGaloisConj: %d hops on %Z tests
GaloisConj:Testing %ZGaloisConj: not found, %d hops 
Best lift: %d
galoisconj _may_ hang up for this polynomialGaloisConj:next p=%ld
GaloisConj:Orbite:%Z
GaloisConj:Frobenius:%Z
GaloisConj: Fixed field %Z
GaloisConj:increase prec of p-adic roots of %ld.
GaloisConj:Back to Earth:%Z
GaloisConj: G[%d]=%Z of relative order %d
GaloisConj: B=%Z
GaloisConj: exp %d: s=%ld [%ld] a=%ld w=%ld wg=%ld sr=%ld
Combinatorics too hard : would need %Z tests!
 I'll skip it but you will get a partial result...%d%% testpermutation(%Z)GaloisConj:%d hop sur %Z iterations
GaloisConj:Fini!
polynomial not in Z[X] in galoisconj4non-monic polynomial in galoisconj4Second arg. must be integer in galoisconj4galoisborne()rootpadicfast()vandermondeinversemod()GaloisConj:%Z
%d Calcul polynomesNumberOfConjugates:Nbtest=%ld,card=%ld,p=%ld
NumberOfConjugates:card=%ld,p=%ld
conjugates list may be incomplete in nfgaloisconjplease apply galoisinit firstNot a Galois field in a Galois related functionfield not Galois or Galois group not weakly super solvableGaloisFixedField:cosets=%Z 
GaloisFixedField:den=%Z mod=%Z 
priority of optional variable too high in galoisfixedfield$08<HKP`lx6,@this function has been suppressedO(a^b)=o(a^b)=p-adic or power series zero with precision given by babs(x)=absolute value (or modulus) of xacos(x)=inverse cosine of xacosh(x)=inverse hyperbolic cosine of xaddelladdell(e,z1,z2)=sum of the points z1 and z2 on elliptic curve eaddprimes(x)=add primes in the vector x (with at most 20 components) to the prime tableadj(x)=adjoint matrix of xagm(x,y)=arithmetic-geometric mean of x and yakell(e,n)=computes the n-th Fourier coefficient of the L-function of the elliptic curve ealgdep(x,n)=algebraic relations up to degree n of xalgdep2GLLpalgdep2(x,n,dec)=algebraic relations up to degree n of x where dec is as in  lindep2algtobasis(nf,x)=transforms the algebraic number x into a column vector on the integral basis nf[7]anellanell(e,n)=computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<32768)apell(e,p)=computes a_p for the elliptic curve e using Shanks-Mestre's methodapell2apell2(e,p)=computes a_p for the elliptic curve e using Jacobi symbolsapprpadicapprpadic(x,a)=p-adic roots of the polynomial x congruent to a mod parg(x)=argument of x,such that -pi<arg(x)<=piasin(x)=inverse sine of xasinh(x)=inverse hyperbolic sine of xassmat(x)=associated matrix to polynomial xatan(x)=inverse tangent of xatanh(x)=inverse hyperbolic tangent of xbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithmbasis2basis2(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 2 algorithmbasistoalg(nf,x)=transforms the vertical vector x on the integral basis into an algebraic numberbernreal(x)=Bernoulli number B_x, as a real number with the current precisionbernvec(x)=Vector of rational Bernoulli numbers B_0, B_2,... up to B_(2x)bestappr(x,k)=gives the best approximation to the real x with denominator less or equal to kbezout(x,y)=gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dbezoutres(x,y)=gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbigomega(x)=number of repeated prime divisors of xbilhell(e,z1,z2)=canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsbin(x,y)=binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbinary(x)=gives the vector formed by the binary digits of x (x C-integer)bittest(x,n)=gives bit number n (coefficient of 2^n) of the integer xboundcfboundcf(x,lmax)=continued fraction expansion of x with at most lmax termsboundfactboundfact(x,lim)=partial factorization of the integer x (using primes up to lim)buchcertifybuchcertify(bnf)=certify the correctness (i.e. remove the GRH) of the bnf data output by buchinit or buchinitfubuchfubuchfu(bnf)=compute the fundamental units of the number field bnf output by buchinitbuchgenGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,pbuchgen(P,...)=compute the structure of the class group and the regulator for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)buchgenforcefubuchgenforcefu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units for the number field defined by the polynomial P, and insist until the units are obtained. See manual for the other parameters (which can be omitted)buchgenfubuchgenfu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units (if they are not too large) for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)buchimagGD0.1,G,D0.1,G,D5,G,buchimag(D,...)=compute the structure of the class group of the complex quadratic field of discriminant D<0. See manual for the other parameters (which can be omitted)buchinit(P,...)=compute the necessary data for future use in ideal and unit group computations. See manual for detailsbuchinitforcefubuchinitforcefu(P,...)=compute the necessary data for future use in ideal and unit group computations, and insist on having fundamental units. See manual for detailsbuchinitfu(P,...)=compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. See manual for detailsbuchnarrowbuchnarrow(bnf)=given a big number field as output by buchinitxx, gives as a 3-component vector the structure of the narrow class groupbuchraybuchray(bnf,ideal)=given a big number field as output by buchinitfu (only) and  an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this modulebuchrayinitbuchrayinit(bnf,ideal)=same as buchrayinitgen, except that the generators are not explicitly computedbuchrayinitgenbuchrayinitgen(bnf,ideal)=given a big number field as output by buchinitfu (only) and  an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data for computing in the ray class group  corresponding to this module. In particular, the fifth component is the ray class group structurebuchrealGD0,G,D0.1,G,D0.1,G,D5,G,pbuchreal(D,...)=compute the structure of the class group and the regulator of the real quadratic field of discriminant D>0 in the wide sense. See manual for the other parameters (which can be omitted)bytesizebytesize(x)=number of bytes occupied by the complete tree of the object xceil(x)=ceiling of x=smallest integer>=xcenterlift(x)=centered lift of x. Same as lift except for integermodscf(x)=continued fraction expansion of x (x rational,real or rational function)cf2(b,x)=continued fraction expansion of x (x rational,real or rational function), where b is the vector of numerators of the continued fractionchangevar(x,y)=change variables of x according to the vector yGnchar(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using the comatrixchar1char1(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using Lagrange interpolationchar2char2(x,y)=characteristic polynomial of the matrix x expressed with variable y, using the Hessenberg form. Can be much faster or much slower than char, depending on the base ringchellchell(x,y)=change data on elliptic curve according to y=[u,r,s,t]chinese(x,y)=x,y being integers modulo mx and my,finds z such that z is congruent to x mod mx and y mod mychptellchptell(x,y)=change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]classno(x)=class number of discriminant xclassno2(x)=class number of discriminant xcoeff(x,s)=coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[])compimagcompimag(x,y)=Gaussian composition of the binary quadratic forms x and y of negative discriminantcompo(x,s)=the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]compositum(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2compositum2compositum2(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2, with roots of pol1 and pol2 expressed on the compositum polynomialscomprealrawcomprealraw(x,y)=Gaussian composition without reduction of the binary quadratic forms x and y of positive discriminantconcat(x,y)=concatenation of x and yGDGDGD1,G,conductor(bnr,subgroup)=conductor of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupconductorofchar(bnr,chi)=conductor of the character chi on the ray class group bnrconj(x)=the algebraic conjugate of xconjvec(x)=conjugate vector of the algebraic number xcontent(x)=gcd of all the components of x, when this makes senseconvol(x,y)=convolution (or Hadamard product) of two power seriescore(n)=unique (positive of negative) squarefree integer d dividing n such that n/d is a squarecore2core2(n)=(long)gen_2-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecoredisc(n)=discriminant of the quadratic field Q(sqrt(n))coredisc2coredisc2(n)=(long)gen_2-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integercos(x)=cosine of xcosh(x)=hyperbolic cosine of xcvtoicvtoi(x)=truncation of x, without taking into account loss of integer part precisioncyclo(n)=n-th cyclotomic polynomialdecodefactordecodefactor(fa)=given a factorisation fa, gives the factored object backdecodemodule(nf,fa)=given a coded module fa as in discrayabslist, gives the true moduledegree(x)=degree of the polynomial or rational function x. -1 if equal 0, 0 if non-zero scalardenom(x)=denominator of x (or lowest common denominator in case of an array)deplin(x)=finds a linear dependence between the columns of the matrix xderiv(x,y)=derivative of x with respect to the main variable of ydet(x)=determinant of the matrix xdet2(x)=determinant of the matrix x (better for integer entries)detint(x)=some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with hermitemoddiagonal(x)=creates the diagonal matrix whose diagonal entries are the entries of the vector xdilog(x)=dilogarithm of xdirdiv(x,y)=division of the Dirichlet series x by the Dir. series yV=GGIDGdireuler(p=a,b,expr)=Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s)dirmul(x,y)=multiplication of the Dirichlet series x by the Dir. series ydirzetak(nf,b)=Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1disc(x)=discriminant of the polynomial xdiscf(x)=discriminant of the number field defined by the polynomial x using round 4discf2discf2(x)=discriminant of the number field defined by the polynomial x using round 2discrayabsGD0,G,D0,G,D0,L,discrayabs(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupGD0,G,D0,G,D2,L,discrayabscond(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if fmodule is not the conductordiscrayabslist(bnf,listes)=if listes is a 2-component vector as output by ideallistunit or similar, gives list of corresponding discrayabsconddiscrayabslistarch(bnf,arch,bound)=gives list of discrayabscond of all modules up to norm bound with archimedean places arch, in a longvector formatdiscrayabslistarchalldiscrayabslistarchall(bnf,bound)=gives list of discrayabscond of all modules up to norm bound with all possible archimedean places arch in reverse lexicographic order, in a longvector formatdiscrayabslistlongdiscrayabslistlong(bnf,bound)=gives list of discrayabscond of all modules up to norm bound without archimedean places, in a longvector formatdiscrayrelGD0,G,D0,G,D1,L,discrayrel(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayrelcondGD0,G,D0,G,D3,L,discrayrelcond(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if module is not the conductordivisors(x)=gives a vector formed by the divisors of x in increasing orderdivres(x,y)=euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainderdivsumdivsum(n,X,expr)=sum of expression expr, X running over the divisors of neigen(x)=eigenvectors of the matrix x given as columns of a matrixeint1(x)=exponential integral E1(x)erfc(x)=complementary error functioneta(x)=eta function without the q^(1/24)euler=euler()=euler's constant with current precisioneval(x)=evaluation of x, replacing variables by their valueexp(x)=exponential of xextract(x,y)=extraction of the components of the vector x according to the vector or mask y, from left to right (1, 2, 4, 8, ...for the first, second, third, fourth,...component)fact(x)=factorial of x (x C-integer), the result being given as a real numberfactcantorfactcantor(x,p)=factorization mod p of the polynomial x using Cantor-Zassenhausfactfqfactfq(x,p,a)=factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factmod(x,p)=factorization mod p of the polynomial x using Berlekampfactor(x)=factorization of xfactoredbasisGGffactoredbasis(x,p)=integral basis of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoreddiscffactoreddiscf(x,p)=discriminant of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoredpolredfactoredpolred(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives minimal polynomials only)factoredpolred2factoredpolred2(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives elements and minimal polynomials)factornf(x,t)=factorization of the polynomial x over the number field defined by the polynomial tfactorpadic(x,p,r)=p-adic factorization of the polynomial x to precision r, using the round 4 algorithmfactorpadic2factorpadic2(x,p,r)=p-adic factorization of the polynomial x to precision r, using Buchmann-LenstraGLLfactpol(x,l,hint)=factorization over Z of the polynomial x up to degree l (complete if l=0) using Hensel lift, knowing that the degree of each factor is a multiple of hintfactpol2factpol2(x,l)=factorization over Z of the polynomial x up to degree l (complete if l=0) using root findingfibofibo(x)=fibonacci number of index x (x C-integer)floor(x)=floor of x=largest integer<=xfor(X=a,b,seq)=the sequence is evaluated, X going from a up to bfordiv(n,X,seq)=the sequence is evaluated, X running over the divisors of nforprime(X=a,b,seq)=the sequence is evaluated, X running over the primes between a and bforstep(X=a,b,s,seq)=the sequence is evaluated, X going from a to b in steps of sforvec(x=v,seq)=v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1fpnfpn(p,n)=monic irreducible polynomial of degree n over F_p[x]frac(x)=fractional part of x=x-floor(x)galois(x)=Galois group of the polynomial x (see manual for group coding)galoisapply(nf,aut,x)=Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfgaloisconj(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field, using p-adics, LLL on integral basis (not always complete)galoisconj1galoisconj1(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field nf, using complex numbers, LLL on integral basis (not always complete)galoisconjforcegaloisconjforce(nf)=list of conjugates of a root of the polynomial x=nf[1] in the Galois number field nf, using p-adics, LLL on integral basis. Guaranteed to be complete if the field is Galois, otherwise there is an infinite loopgamhgamh(x)=gamma of x+1/2 (x integer)gamma(x)=gamma function at xgauss(a,b)=gaussian solution of ax=b (a matrix,b vector)gaussmodulogaussmodulo(M,D,Y)=(long)gen_1 solution of system of congruences MX=Y mod Dgaussmodulo2gaussmodulo2(M,D,Y)=all solutions of system of congruences MX=Y mod Dgcd(x,y)=greatest common divisor of x and ygetheap()=2-component vector giving the current number of objects in the heap and the space they occupygetrand()=current value of random number seedgetstack()=current value of stack pointer avmagettime()=time (in milliseconds) since last call to gettimeglobalred(e)=e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pgotogoto(n)=THIS FUNCTION HAS BEEN SUPPRESSEDhclassno(x)=Hurwitz-Kronecker class number of x>0hell(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using theta-functionshell2hell2(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using Tate's methodhermite(x)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, using a naive algorithmhermite2(x)=2-component vector [H,U] such that H is an (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, and U is a unimodular matrix such that xU=H, using Batut's algorithmhermitehavashermitehavas(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Havas's algorithmhermitemod(x,d)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is the non-zero determinant of this latticehermitemodidhermitemodid(x,d)=(upper triangular) Hermite normal form of x concatenated with d times the identity matrixhermitepermhermiteperm(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Batut's algorithmhess(x)=Hessenberg form of xhilbhilb(x,y,p)=Hilbert symbol at p of x,y (integers or fractions)hilbert(n)=Hilbert matrix of order n (n C-integer)hilbphilbp(x,y)=Hilbert symbol of x,y (where x or y is integermod or p-adic)hvectorhvector(n,X,expr)=row vector with n components of expression expr, the variable X ranging from 1 to nhyperu(a,b,x)=U-confluent hypergeometric functioni=i()=square root of -1idealadd(nf,x,y)=sum of two ideals x and y in the number field defined by nfidealaddmultoneidealaddone(nf,x,y)=when the sum of two ideals x and y in the number field K defined by nf is equal to Z_K, gives a two-component vector [a,b] such that a is in x, b is in y and a+b=1idealaddoneidealaddmultone(nf,list)=when the sum of the ideals in the number field K defined by nf and given in the vector list is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1idealappr(nf,x)=x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealapprfact(nf,x)=x being a prime ideal factorization with possibly zero or negative exponents, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealchinese(nf,x,y)=x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealcoprime(nf,x,y)=gives an element b in nf such that b.x is an integral ideal coprime to the integral ideal yidealdiv(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nfidealdivexact(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nf when the quotient is known to be an integral idealidealfactor(nf,x)=factorization of the ideal x given in HNF into prime ideals in the number field nfidealhermite(nf,x)=hermite normal form of the ideal x in the number field nf, whatever form x may haveidealhermite2idealhermite2(nf,a,b)=hermite normal form of the ideal aZ_K+bZ_K in the number field K defined by nf, where a and b are elementsidealintersect(nf,x,y)=intersection of two ideals x and y in HNF in the number field defined by nfidealinv(nf,x)=inverse of the ideal x in the number field nf not using the differentidealinv2idealinv2(nf,x)=inverse of the ideal x in the number field nf using the differentideallist(nf,bound)=vector of vectors of all ideals of norm<=bound in nfideallistarch(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, without generatorsideallistarchgenideallistarchgen(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, with generatorsideallistunit(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, without generatorsideallistunitarchideallistunitarch(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitideallistunitarchgenideallistunitarchgen(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitgenideallistunitgen(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, with generatorsideallistzstarideallistzstar(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, without generatorsideallistzstargenideallistzstargen(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, with generatorsideallllredideallllred(nf,x,vdir)=LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealmul(nf,x,y)=product of the two ideals x and y in the number field nfidealmulredidealmulred(nf,x,y)=reduced product of the two ideals x and y in the number field nfidealnorm(nf,x)=norm of the ideal x in the number field nfidealpow(nf,x,n)=n-th power of the ideal x in HNF in the number field nfidealpowred(nf,x,n)=reduced n-th power of the ideal x in HNF in the number field nfidealtwoelt(nf,x)=(long)gen_2-element representation of an ideal x in the number field nfidealtwoelt2idealtwoelt2(nf,x,a)=(long)gen_2-element representation of an ideal x in the number field nf, with the first element equal to aidealval(nf,x,p)=valuation at p given in primedec format of the ideal x in the number field nfidmatidmat(n)=identity matrix of order n (n C-integer)if(a,seq1,seq2)=if a is nonzero, seq1 is evaluated, otherwise seq2imag(x)=imaginary part of ximage(x)=basis of the image of the matrix ximage2image2(x)=basis of the image of the matrix ximagecompl(x)=vector of column indices not corresponding to the indices given by the function imageincgam(s,x)=incomplete gamma functionincgam1incgam1(s,x)=incomplete gamma function (for debugging only)incgam2(s,x)=incomplete gamma function (for debugging only)incgam3incgam3(s,x)=complementary incomplete gamma functionincgam4incgam4(s,x,y)=incomplete gamma function where y=gamma(s) is precomputedindexrank(x)=gives two extraction vectors (rows and columns) for the matrix x such that the exracted matrix is square of maximal rankindsortindsort(x)=indirect sorting of the vector xinitalg(x)=x being a nonconstant irreducible polynomial, gives the vector: [x,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]initalgredinitalgred(x)=x being a nonconstant irreducible polynomial, finds (using polred) a simpler polynomial pol defining the same number field, and gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual), r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]initalgred2initalgred2(P)=P being a nonconstant irreducible polynomial, gives a two-element vector [nf,mod(a,pol)], where nf is as output by initalgred and mod(a,pol) is a polymod equal to mod(x,P) and pol=nf[1]initell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j,[e1,e2,e3],w1,w2,eta1,eta2,q,area]initzetainitzeta(x)=compute number field information necessary to use zetak, where x is an irreducible polynomialinteg(x,y)=formal integration of x with respect to the main variable of yintersect(x,y)=intersection of the vector spaces whose bases are the columns of x and yintgenV=GGID1,L,pintgen(X=a,b,s)=general numerical integration of s from a to b with respect to X, to be used after removing singularitiesintinfV=GGID2,L,pintinf(X=a,b,s)=numerical integration of s from a to b with respect to X, where a or b can be plus or minus infinity (1.0e4000), but of same signV=GGID0,L,pintnum(X=a,b,s)=numerical integration of s from a to b with respect to XintopenV=GGID3,L,pintopen(X=a,b,s)=numerical integration of s from a to b with respect to X, where s has only limits at a or binverseimage(x,y)=an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwiseisdiagonal(x)=true(1) if x is a diagonal matrix, false(0) otherwiseisfundisfund(x)=true(1) if x is a fundamental discriminant (including 1), false(0) if notisideal(nf,x)=true(1) if x is an ideal in the number field nf, false(0) if notisincl(x,y)=tests whether the number field defined by the polynomial x is isomorphic to a subfield of the one defined by y; 0 if not, otherwise all the isomorphismsisinclfastisinclfast(nf1,nf2)=tests whether the number nf1 is isomorphic to a subfield of nf2 or not. If it gives a non-zero result, this proves that this is the case. However if it gives zero, nf1 may still be isomorphic to a subfield of nf2 so you have to use the much slower isincl to be sureisirreducible(x)=true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantisisom(x,y)=tests whether the number field defined by the polynomial x is isomorphic to the one defined by y; 0 if not, otherwise all the isomorphismsisisomfastisisomfast(nf1,nf2)=tests whether the number fields nf1 and nf2 are isomorphic or not. If it gives a non-zero result, this proves that they are isomorphic. However if it gives zero, nf1 and nf2 may still be isomorphic so you have to use the much slower isisom to be sureisoncurve(e,x)=true(1) if x is on elliptic curve e, false(0) if notisprime(x)=true(1) if x is a strong pseudoprime for 10 random bases, false(0) if notisprincipal(bnf,x)=bnf being output by buchinit, gives the vector of exponents on the class group generators of x. In particular x is principal if and only if the result is the zero vectorisprincipalforceisprincipalforce(bnf,x)=same as isprincipal, except that the precision is doubled until the result is obtainedisprincipalgenisprincipalgen(bnf,x)=bnf being output by buchinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorisprincipalgenforceisprincipalgenforce(bnf,x)=same as isprincipalgen, except that the precision is doubled until the result is obtainedisprincipalray(bnf,x)=bnf being output by buchrayinit, gives the vector of exponents on the ray class group generators of x. In particular x is principal if and only if the result is the zero vectorisprincipalraygenisprincipalraygen(bnf,x)=bnf being output by buchrayinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorispspispsp(x)=true(1) if x is a strong pseudoprime, false(0) if notisqrtisqrt(x)=integer square root of x (x integer)isset(x)=true(1) if x is a set (row vector with strictly increasing entries), false(0) if notissqfreeissqfree(x)=true(1) if x is squarefree, false(0) if notissquare(x)=true(1) if x is a square, false(0) if notisunit(bnf,x)=bnf being output by buchinit, gives the vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisejacobi(x)=eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xjbesselh(n,x)=J-bessel function of index n+1/2 and argument x, where n is a non-negative integerjelljell(x)=elliptic j invariant of xkaramulkaramul(x,y,k)=THIS FUNCTION HAS BEEN SUPPRESSEDkbessel(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)kbessel2kbessel2(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)ker(x)=basis of the kernel of the matrix xkerikeri(x)=basis of the kernel of the matrix x with integer entrieskerint(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkerint1kerint1(x)=LLL-reduced Z-basis of the kernel of the matrix x with rational entries using matrixqz3 and the HNFkerint2kerint2(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkrokro(x,y)=kronecker symbol (x/y)labellabel(n)=THIS FUNCTION HAS BEEN SUPPRESSEDlambdaklambdak(nfz,s)=Dedekind lambda function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)laplace(x)=replaces the power series sum of a_n*x^n/n! by sum of a_n*x^nlcm(x,y)=least common multiple of x and y=x*y/gcd(x,y)legendre(n)=legendre polynomial of degree n (n C-integer)length(x)=number of non code words in xlex(x,y)=compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)lexsort(x)=sort the elements of the vector x in ascending lexicographic orderlift(x)=lifts every element of Z/nZ to Z or Z[x]/PZ[x] to Z[x]lindep(x)=Z-linear dependencies between components of x (Hastad et al)lindep2(x,dec)=Z-linear dependencies between components of x using LLL, where dec should be about one half the number of decimal digits of precisionlll(x)=lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lll1lll1(x)=old version of lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lllgenlllgen(x)=lll reduction of the vectors forming the matrix x with polynomial coefficients (gives the unimodular transformation matrix)lllgram(x)=lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)lllgram1lllgram1(x)=old version of lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)lllgramgenlllgramgen(x)=lll reduction of the lattice whose gram matrix is x with polynomial coefficients (gives the unimodular transformation matrix)lllgramintlllgramint(x)=lll reduction of the lattice whose gram matrix is the integral matrix x (gives the unimodular transformation matrix)lllgramkerimlllgramkerim(x)=kernel and lll reduction of the lattice whose gram matrix is the integral matrix xlllgramkerimgenlllgramkerimgen(x)=kernel and lll reduction of the lattice whose gram matrix is the matrix x with polynomial coefficientslllint(x)=lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllintpartial(x)=partial (hence faster) lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllkerimlllkerim(x)=kernel and lll reduction of the vectors forming the integral matrix xlllkerimgenlllkerimgen(x)=kernel and lll reduction of the vectors forming the matrix x with polynomial coefficientslllratlllrat(x)=lll reduction of the vectors forming the matrix x, computations done with rational numbers (gives the unimodular transformation matrix)ln(x)=log(x)=natural logarithm of xlngamma(x)=logarithm of the gamma function of xlocalred(e,p)=e being an ellliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_plog(x)=ln(x)=natural logarithm of xlogagmlogagm(x)=natural logarithm of x, computed using agm (faster than log for more than a few hundred decimal digits)GGGGplseriesell(e,s,N,A)=L-series at s of the elliptic curve e, where |N| is the conductor, sign(N) the sign of the functional equation, and A a cut-off point close to 1makebigbnf(sbnf)=transforms small sbnf as output by smallbuchinit into a true big bnfmat(x)=transforms any GEN x into a matrixmatextractmatextract(x,y,z)=extraction of the components of the matrix x according to the vector or masks y (for the rows) and z (for the columns) from left to right (1,2,4,8,...for the first, second, third, fourth, ...rows or columns)mathellmathell(e,x)=gives the height matrix for vector of points x on elliptic curve e using theta functionsGGVVImatrix(m,n,X,Y,expr)=mXn matrix of expression expr, the row variable X going  from 1 to m and the column variable Y going from 1 to nmatrixqz(x,p)=transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwisematrixqz2(x)=finds a basis of the intersection with Z^n of the lattice spanned by the columns of xmatrixqz3(x)=finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsize(x)=number of rows and columns of the vector/matrix x as a 2-vectormax(x,y)=maximum of x and ymin(x,y)=minimum of x and yminidealminideal(nf,ix,vdir)=minimum of the ideal ix in the direction vdir in the number field nfminim(x,bound,maxnum)=number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0minim2(x,bound)=looks for vectors of square norm <= bound, return the first one and its normmod(x,y)=creates the integer x modulo y on the PARI stackmodpmodp(x,y)=creates the integer x modulo y as a permanent object (on the heap)modreverse(x)=reverse polymod of the polymod x, if it existsmodulargcd(x,y)=gcd of the polynomials x and y using the modular methodmumu(x)=Moebius function of xnewtonpoly(x,p)=Newton polygon of polynomial x with respect to the prime pnextprime(x)=smallest prime number>=xnfdetint(nf,x)=multiple of the ideal determinant of the pseudo generating set xnfdiv(nf,a,b)=element a/b in nfnfdiveucnfdiveuc(nf,a,b)=gives algebraic integer q such that a-bq is smallnfdivresnfdivres(nf,a,b)=gives [q,r] such that r=a-bq is smallnfhermite(nf,x)=if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhermitemodnfhermitemod(nf,x,detx)=if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfmod(nf,a,b)=gives r such that r=a-bq is small with q algebraic integernfmul(nf,a,b)=element a.b in nfnfpow(nf,a,k)=element a^k in nfnfreducenfreduce(nf,a,id)=gives r such that a-r is the ideal id and r is smallnfsmith(nf,x)=if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfvalnfval(nf,a,pr)=valuation of element a at the prime prnorm(x)=norm of xnorml2(x)=square of the L2-norm of the vector xnucomp(x,y,l)=composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputednumdiv(x)=number of divisors of xnumer(x)=numerator of xnupow(x,n)=n-th power of primitive positive definite quadratic form x using nucomp and nuduplo(a^b)=O(a^b)=p-adic or power series zero with precision given by bomega(x)=number of unrepeated prime divisors of xordellordell(e,x)=y-coordinates corresponding to x-ordinate x on elliptic curve eorder(x)=order of the integermod x in (Z/nZ)*orderellorderell(e,p)=order of the point p on the elliptic curve e over Q, 0 if non-torsionordredordred(x)=reduction of the polynomial x, staying in the same orderpadicprec(x,p)=absolute p-adic precision of object xpascal(n)=pascal triangle of order n (n C-integer)perfperf(a)=rank of matrix of xx~ for x minimal vectors of a gram matrix apermutationpermutation(n,k)=permutation number k (mod n!) of n letters (n C-integer)permutation2numpermutation2num(vect)=ordinal (between 1 and n!) of permutation vectpfpf(x,p)=returns the prime form whose first coefficient is p, of discriminant xphi(x)=Euler's totient function of xpi=pi()=the constant pi, with current precisionpnqn(x)=[p_n,p_{n-1};q_n,q_{n-1}] corresponding to the continued fraction xpointellpointell(e,z)=coordinates of point on the curve e corresponding to the complex number zGGGD&polint(xa,ya,x)=polynomial interpolation at x according to data vectors xa, yapolred(x)=reduction of the polynomial x (gives minimal polynomials only)polred2(x)=reduction of the polynomial x (gives elements and minimal polynomials)polredabs(x)=a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 normpolredabs2polredabs2(x)=gives [pol,a] where pol is as in polredabs, and alpha is the element whose characteristic polynomial is polpolredabsallpolredabsall(x)=complete list of the smallest generating polynomials of the number field for the T2 norm on the rootspolredabsfastpolredabsfast(x)=a smallest generating polynomial of the number field for the T2 norm on the rootspolredabsnoredpolredabsnored(x)=a smallest generating polynomial of the number field for the T2 norm on the roots without initial polredpolsym(x,n)=vector of symmetric powers of the roots of x up to npolvar(x)=main variable of object x. Gives p for p-adic x, error for scalarspoly(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientLGppolylog(m,x)=m-th polylogarithm of xpolylogdpolylogd(m,x)=D_m~-modified m-th polylog of xpolylogdoldpolylogdold(m,x)=D_m-modified m-th polylog of xpolylogppolylogp(m,x)=P_m-modified m-th polylog of xpolyrevpolyrev(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termpolzagpolzag(n,m)=Zagier's polynomials of index n,mpowell(e,x,n)=n times the point x on elliptic curve e (n in Z)powrealraw(x,n)=n-th power without reduction of the binary quadratic form x of positive discriminantprec(x,n)=change the precision of x to be n (n C-integer)precision(x)=real precision of object xprime(n)=returns the n-th prime (n C-integer)primedec(nf,p)=prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a.Z_K, e,f as usual, a as vector of components on the  integral basis, b Lenstra's constantprimes(n)=returns the vector of the first n primes (n C-integer)primroot(n)=returns a primitive root of n when it existsprincipalidealprincipalideal(nf,x)=returns the principal ideal generated by the algebraic number x in the number field nfprincipalideleprincipalidele(nf,x)=returns the principal idele generated by the algebraic number x in the number field nfGV=GGIprod(x,X=a,b,expr)=x times the product (X runs from a to b) of expressionV=GGIpprodeuler(X=a,b,expr)=Euler product (X runs over the primes between a and b) of real or complex expressionV=GID0,L,pprodinf(X=a,expr)=infinite product (X goes from a to infinity) of real or complex expressionV=GID1,L,pprodinf1(X=a,expr)=infinite product (X goes from a to infinity) of real or complex 1+expressionpsi(x)=psi-function at xqfiqfi(a,b,c)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c<0qfr(a,b,c,d)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c>0 and distance dquaddisc(x)=discriminant of the quadratic field Q(sqrt(x))quadgen(x)=standard generator of quadratic order of discriminant xquadpoly(x)=quadratic polynomial corresponding to the discriminant xrandom()=random integer between 0 and 2^31-1rank(x)=rank of the matrix xrayclassnorayclassno(bnf,x)=ray class number of the module x for the big number field bnf. Faster than buchray if only the ray class number is wantedrayclassnolistrayclassnolist(bnf,liste)=if listes is as output by idealisunit or similar, gives list of corresponding ray class numbersreal(x)=real part of xrecip(x)=reciprocal polynomial of xredimag(x)=reduction of the binary quadratic form x with D<0redreal(x)=reduction of the binary quadratic form x with D>0redrealnodredrealnod(x,sq)=reduction of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]reduceddiscreduceddisc(f)=vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fregula(x)=regulator of the real quadratic field of discriminant xreorder(x)=reorder the variables for output according to the vector xresultant(x,y)=resultant of the polynomials x and y with exact entriesresultant2resultant2(x,y)=resultant of the polynomials x and yreverse(x)=reversion of the power series xrhorealrhoreal(x)=single reduction step of the binary quadratic form x of positive discriminantrhorealnodrhorealnod(x,sq)=single reduction step of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]rndtoi(x)=take the nearest integer to all the coefficients of x, without taking into account loss of integer part precisionrnfbasis(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setrnfdiscfrnfdiscf(nf,pol)=given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfequation(nf,pol)=given a pol with coefficients in nf, gives the absolute equation of the number field defined by polrnfequation2rnfequation2(nf,pol)=given a pol with coefficients in nf, gives [apol,th], where apol is the absolute equation of the number field defined by pol and th expresses the root of nf[1] in terms of the root of apolrnfhermitebasisrnfhermitebasis(bnf,order)=given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfisfree(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notrnflllgram(nf,pol,order)=given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixrnfpolred(nf,pol)=given a pol with coefficients in nf, finds a list of polynomials defining some subfields, hopefully simplerrnfpseudobasis(nf,pol)=given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfsteinitz(nf,order)=given an order as output by rnfpseudobasis, gives [A,I,..] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialrootmod(x,p)=roots mod p of the polynomial xrootmod2(x,p)=roots mod p of the polynomial x, when p is smallrootpadic(x,p,r)=p-adic roots of the polynomial x to precision rroots(x)=roots of the polynomial x using Schonhage's method modified by Gourdonrootsof1(nf)=number of roots of unity and primitive root of unity in the number field nfrootsoldrootsold(x)=roots of the polynomial x using a modified Newton's methodround(x)=take the nearest integer to all the coefficients of xrounderrorrounderror(x)=maximum error found in rounding xseries(x,v)=convert x (usually a vector) into a power series with variable v, starting with the constant coefficientset(x)=convert x into a set, i.e. a row vector with strictly increasing coefficientssetintersect(x,y)=intersection of the sets x and ysetminus(x,y)=set of elements of x not belonging to ysetrand(n)=reset the seed of the random number generator to nsetsearch(x,y)=looks if y belongs to the set x. Returns 0 if it is not, otherwise returns the index j such that y==x[j]setunion(x,y)=union of the sets x and yshift(x,n)=shift x left n bits if n>=0, right -n bits if n<0shiftmul(x,n)=multiply x by 2^n (n>=0 or n<0)sigma(x)=sum of the divisors of xsigmaksigmak(k,x)=sum of the k-th powers of the divisors of x (k C-integer)sign(x)=sign of x, of type integer, real or fractionsignatsignat(x)=signature of the symmetric matrix xsignunit(bnf)=matrix of signs of the real embeddings of the system of fundamental units found by buchinitsimplefactmodsimplefactmod(x,p)=same as factmod except that only the degrees of the irreducible factors are givensimplify(x)=simplify the object x as much as possiblesin(x)=sine of xsinh(x)=hyperbolic sine of xsize(x)=maximum number of decimal digits minus one of (the coefficients of) xsmallbasissmallbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallbuchinitsmallbuchinit(pol)=small buchinit, which can be converted to a big one using makebigbnfsmalldiscfsmalldiscf(x)=discriminant of the number field defined by the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallfactsmallfact(x)=partial factorization of the integer x (using only the stored primes)smallinitellsmallinitell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j]smallpolredsmallpolred(x)=partial reduction of the polynomial x (gives minimal polynomials only)smallpolred2smallpolred2(x)=partial reduction of the polynomial x (gives elements and minimal polynomials)smith(x)=Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vectorsmith2smith2(x)=gives a three element vector [u,v,d] where u and v are square unimodular matrices such that d=u*x*v=diagonal(smith(x))smithclean(z)=if z=[u,v,d] as output by smith2, removes from u,v,d the rows and columns corresponding to entries equal to 1 in dsmithpolsmithpol(x)=Smith normal form (i.e. elementary divisors) of the matrix x with polynomial coefficients, expressed as a vectorsolve(X=a,b,expr)=real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sort(x)=sort in ascending order of the vector xsqr(x)=square of x. NOT identical to x*xsqredsqred(x)=square reduction of the (symmetric) matrix x ( returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)sqrt(x)=square root of xsrgcd(x,y)=polynomial gcd of x and y using the subresultant algorithmsturm(x)=number of real roots of the polynomial xsturmpartsturmpart(x,a,b)=number of real roots of the polynomial x in the interval (a,b]subcyclo(p,d)=finds an equation for the d-th degree subfield of Q(zeta_p), where p must be a prime powersubellsubell(e,z1,z2)=difference of the points z1 and z2 on elliptic curve esubst(x,y,z)=in expression x, replace the variable y by the expression zsum(x,X=a,b,expr)=x plus the sum (X goes from a to b) of expression exprsumalt(X=a,expr)=Villegas-Zagier's acceleration of alternating series expr, X starting at asumalt2sumalt2(X=a,expr)=Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at asuminf(X=a,expr)=infinite sum (X goes from a to infinity) of real or complex expression exprsumpos(X=a,expr)=sum of positive series expr, the formal variable X starting at asumpos2(X=a,expr)=sum of positive series expr, the formal variable X starting at a, using Zagier's polynomialssupplement(x)=supplement the columns of the matrix x to an invertible matrixsylvestermatrix(x,y)=forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowstan(x)=tangent of xtanh(x)=hyperbolic tangent of xtaniyama(e)=modular parametrization of elliptic curve etaylor(x,y)=taylor expansion of x with respect to the main variable of ytchebi(n)=Tchebitcheff polynomial of degree n (n C-integer)teichteich(x)=teichmuller character of p-adic number xtheta(q,z)=Jacobi sine theta-functionthetanullk(q,k)=k'th derivative at z=0 of theta(q,z)threetotwothreetotwo(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]threetotwo2threetotwo2(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]torsell(e)=torsion subgroup of elliptic curve e: order, structure, generatorstrace(x)=trace of xtrans(x)=x~=transpose of xtrunc(x)=truncation of x;when x is a power series,take away the O(X^)tschirnhaus(x)=random Tschirnhausen transformation of the polynomial xtwototwotwototwo(nf,a,b)=returns a 3-component vector [d,e,U] such that U is a unimodular 2x2 matrix with algebraic integer coefficients such that [a,b]*U=[d,e] and d,e are hopefully smallerunit(x)=fundamental unit of the quadratic field of discriminant x where x must be positiveuntil(a,seq)=evaluate the expression sequence seq until a is nonzerovaluation(x,p)=valuation of x with respect to pvec(x)=transforms the object x into a vector. Used mainly if x is a polynomial or a power seriesvecindexsortvecindexsort(x): indirect sorting of the vector xveclexsortveclexsort(x): sort the elements of the vector x in ascending lexicographic ordervecmax(x)=maximum of the elements of the vector/matrix xvecmin(x)=minimum of the elements of the vector/matrix xvecsort(x,k)=sorts the vector of vector (or matrix) x according to the value of its k-th componentvector(n,X,expr)=row vector with n components of expression expr (X ranges from 1 to n)vvectorvvector(n,X,expr)=column vector with n components of expression expr (X ranges from 1 to n)weipellweipell(e)=formal expansion in x=z of Weierstrass P functionwfweberf(x)=Weber's f function of x (j=(f^24-16)^3/f^24)wf2weberf2(x)=Weber's f2 function of x (j=(f2^24+16)^3/f2^24)while(a,seq)=while a is nonzero evaluate the expression sequence seq. Otherwise 0zell(e,z)=In the complex case, lattice point corresponding to the point z on the elliptic curve ezeta(s)=Riemann zeta function at szetak(nfz,s)=Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)zideallog(nf,x,bid)=if bid is a big ideal as given by zidealstarinit or zidealstarinitgen , gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)zidealstarzidealstar(nf,I)=3-component vector v, giving the structure of (Z_K/I)^*. v[1] is  the order (i.e. phi(I)), v[2] is a vector of cyclic components, and v[3]  is a vector giving the corresponding generatorszidealstarinitzidealstarinit(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*zidealstarinitgenzidealstarinitgen(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*znstar(n)=3-component vector v, giving the structure of (Z/nZ)^*. v[1] is  the order (i.e. phi(n)), v[2] is a vector of cyclic components, and v[3]  is a vector giving the corresponding generatorsget_archincorrect vector length in idealrednot a vector in idealred0th power in idealpowprime_specincorrect ideal in idealtypzero ideal in idealfactornot a vector of ideals in idealaddmultooneideals don't sum to Z_K in idealaddmultoonegeneric conversion to finite field0 in get_arch_realunif_mod_fZnored + denominator in idealapprfactideal_two_eltcannot invert zero idealincompatible variables in idealinvnon-integral exponent in idealpownon-integral exponent in idealpowredquotient not integral in idealdivexactnot a factorization in idealapprfactnot a prime ideal factorization in idealchinesenot a suitable vector of elements in idealchineseelement not in ideal in ideal_two_elt2element does not belong to ideal in ideal_two_elt2not a module in %snot a matrix in %snot a correct ideal list in %sincorrect idele in idealaddtoonenot a matrix of maximal rank in nfhermitenfhermite, i = %ldnot a module in nfsmithnot a matrix in nfsmithnot a correct ideal list in nfsmithnot a matrix of maximal rank in nfsmithnfsmith for non square matricesbug2 in nfsmith[1]: nfhermitemod, i = %ld[2]: nfhermitemod, i = %ld$bbb$bbbbbCbXbbbbbbbbbhbprimedec: %Z is not primereducible polynomial in allbasedisc. factorisation  entering Dedekind Basis with parameters p=%Z
  f = %Z,
  a = %Z
  new order: %Z
newtonsumsget_normmodpr initialized for integers only!rowred j=%ldnewtonsums: result doesn't fit in cache
  entering Nilord with parameters: %Z^%ld
  fx = %Z, gx = %Z  (Fa, Ea) = (%ld,%ld)
  beta = %Z
  ** switching to normal mode
  ** switching to fast mode
 content in fastnu is %Z
  fastnu: G is computed
  fastnu: HNF(G) is computed
  (eq,er) = (%ld,%ld)
  Increasing Fa
no root in nilord. Is p = %Z a prime?  Increasing Ea
nilord  dedek: gcd has degree %ld
initial parameters p=%Z,
  f=%Z
Treating p^k = %Z^%ld
ROUND2: epsilon = %ld	avma = %ld
impossible inverse: %ZResult for prime %Z is:
%Z
not a factorisation in nfbasis  entering Decomp, parameters: %Z^%ld
  f = %Zbug in Decomp (not a factor), is p = %Z a prime?  leaving Decomp: f1 = %Z
f2 = %Z
e = %Z
de= %Z
IndexPartial: discriminantIndexPartial: factorizationIndexPartial: factor %Z^%ld --> %Z : incorrect modpr formatnf_to_ffincorrect polynomial in rnf functionincorrect coeff in rnf functionnon-monic relative polynomialsIdeals to consider:
 treating %Z
    pass no %ld
 new order:
%Z
%Z
rnfordmaxnot a pseudo-matrix in %snot a pseudo-basis in nfsimplifybasisrnfdet2not a pseudo-matrix in rnfdetpolcompositum0not the same variable in compositumcompositum: %Z inseparableinseparable relative equation in rnfequationthis combination of flags in rnfpolredabsrelative basis computed
absolute basisoriginal absolute generator: %Z
reduced absolute generator: %Z
%3ld %s at prime
  %Z
Time: %ld
exponent: %ld
for this exponent, GSmin = %Z
Time reduction: %ld
polynomial variable must have highest priority in nffactormodnfsqffUsing Trager's method
choice of a prime idealPrime ideal chosen: %Z
nf_factor_boundbound computation  1) T_2 bound for %s: %Z
  2) Conversion from T_2 --> | |^2 bound : %Z
  3) Final bound: %Z
splitting mod %ZHensel liftto find factor %Zfor this tracenf_LLL_cmbf: checking factor %ld (avma - bot = %lu)
... mod p^k (avma - bot = %lu)
... lifted (avma - bot = %lu)
nf_LLL_cmbfpolynomial variable must have highest priority in nfrootstest if polynomial is square-free
nfissplit
Entering nffactor:
polynomial variable must have highest priority in nffactorsquarefree testnumber of factor(s) found: %ld
incorrect variables in rnfcharpoly
tB?get_pol  generator: %Z
checkrnfplease apply bnfinit firstcheckbnfplease apply nfinit firstchecknfincorrect bigray fieldplease apply bnrinit(,,1) and not bnrinit(,)missing units in %sincorrect idealincorrect matrix for idealincorrect bigidealincorrect prime idealpolynomial not in Z[X] in %spolynomial not in Z[X,Y] in %sincompatible modulus in %s:
  mod = %Z,
  nf  = %ZTschirnhaus transform. New pol: %Zgpolcomp (different degrees)S1S2A3S3C(4) = 4E(4) = 2[x]2D(4)A4S4C(5) = 5D(5) = 5:2F(5) = 5:4A5S5C(6) = 6 = 3[x]2D_6(6) = [3]2D(6) = S(3)[x]2A_4(6) = [2^2]3F_18(6) = [3^2]2 = 3 wr 22A_4(6) = [2^3]3 = 2 wr 3S_4(6d) = [2^2]S(3)S_4(6c) = 1/2[2^3]S(3)F_18(6):2 = [1/2.S(3)^2]2F_36(6) = 1/2[S(3)^2]22S_4(6) = [2^3]S(3) = 2 wr S(3)L(6) = PSL(2,5) = A_5(6)F_36(6):2 = [S(3)^2]2 = S(3) wr 2L(6):2 = PGL(2,5) = S_5(6)A6S6C(7) = 7D(7) = 7:2F_21(7) = 7:3F_42(7) = 7:6L(7) = L(3,2)A7S7galois of degree higher than 11galois of reducible polynomialgalois (bug1)galois (bug3)galois (bug2)galois (bug4)incorrect galois automorphism in galoisapplyget_bnfpolmatrix Mnsiso0nfiso or nfinclfalse nf in nf_get_r1false nf in nf_get_r2false nf in nf_get_signget_red_G: starting LLL, prec = %ld (%ld + %ld)
get_red_GLLL basismult. tableround4polred for non-monic polynomialnon-monic polynomial. Result of the form [nf,c]you found a counter-example to a conjecture, please report!xbest = %Z
incorrect nf in nfnewprecchk_gen_init: generator %Z
chk_gen_init: subfield %Z
chk_gen_init: difficult field, trying random elements
precision too low in chk_gen_initchk_gen_init: skipfirst = %ld
chk_gen_init: new prec = %ld (initially %ld)
polredabs (precision problem)polredabs0Found %ld minimal polynomials.
nf_ADDZK flag when nf_ALL set (polredabs)rootsof1 (bug1)not an integer type in dirzetaktoo many terms in dirzetakneed %Z coefficients in initzeta: computation impossible
initzeta: N0 = %Z
i0 = %ld
a(i,j)coefa(n)log(n)Ciknot a zeta number field in zetakallgzetakalls = 1 is a pole (gzetakall)s = 0 is a pole (gzetakall)@{M{{;D]]h]&1]]%]]]]]%	l
<!<!<!<!<@<!<!<!<!<!<!
 z >B>BBBIBPBWB^Q?ɴK?ɴK?x2??zero polynomial in FpXQ_pow. %Z not primespec_FpXQ_powQpX_to_ZXspec_FqXQ_pownot a prime in factmodnot a prime in rootmodeuclidean division (poldivrem)prime too big in rootmod2not a prime in polrootsmodFpXQYQ_powBerlekamp_matrixBerlekamp_kerBerlekamp matrix   %3ld fact. of degree %3ld
   %3ld factor of degree %3ld
FpX_factor_2lx<ly in Flx_addmul_inplacefactmod: %lu is not prime[FqX_split] splitting time: %ld (%ld trials)
FqX_factorFqX_split_Trager: choosing k = %ld
reducible modulus in factornfFqX_split_Trager failed!Zp_apprnon-positive precision in rootpadicnon-positive precision in factorpadicfactorpadic2 for non-monic polynomialpolfnfpolfnf: choosing k = %ld
QpXQ_to_ZXYpolynomial variable must have higher priority in factorffto_Fq_polto_Fq* Finding eigenvalues
too many iterations in hqr* Eigenvalues computed
polynomial has probably multiple roots in zrhqr
polished roots = %Zroots2too many iterations in rootstoo many iterations in roots2() ( laguer() ):
     real coefficients polynomial, using zrhqr()
too many iterations in rootsold(): using roots2()error in rootsold(): using roots2()<3<p=
ף?RQ?ףp=
?)\(?ܿRgX_RgXQ_compoRgX_powersnormalizing a polynomial with 0 leading termzpsolubleqpsolubleqpsolublenfzpsolublenf0 argument in nfhilbertp0 argument in nfhilbertnfhilbert not soluble at real place %ld
nfhilbert not soluble at finite place: %Z
non monic relative equationmain variable must be of higher priority in rnfisnorminitplease apply rnfisnorminit firstuseless flag in rnfisnorm: the extension is Galoisincorrect abscissa in sumnumincorrect table length in intnum initializationincorrect a or b in intnumboth nonzero real and imag. part in coding, real ignoredx = 0 in FourierFourier transform of oscillating functionsqrom2: iteration %ld: %Z
integral from infty to infty or from -infty to -inftycode error in intnumm too large in intnuminitintnuminit0sumnuminit0incorrect beginning value in sumnuminfinities of the same sign in intnuminitgeninfinities of different type in intnuminitgenexponential increase in integral transformneed exponential decrease in intinvmellinshortqrom3: iteration %ld: %Z
KK[[})dܨ^^nWW3333333@please apply rnfequation(,,1)incorrect data in eltreltoabsmain variable must be of higher priority in rnfinitalgelement is not in the base field in rnfelementdownrnfidealhermiteray regulatorray units.furay torsion unitscurve not defined over Rcurve not defined over a p-adic fieldpartition functionarg to partition function must be < 10^15Q?gchsingular argument in atanhgathgashrfix (conversion to t_REAL)caching Bernoulli numbers 2*%ld to 2*%ld, prec = %ld
Bernoulligatangasingachgacoszero argument in garglim, nn: [%ld, %ld], la = %lf
non-positive integer argument in cxgammaproduct from 0 to N-1Bernoullisnon-positive integer in glngammalngamma around a!=1p-adic lngamma functionnon-positive integer argument in ggammaargument too large in ggammaGamma not defined for non-integral p-adic numberargument too large in ggamdgamd of a power seriesnon-positive integer argument in cxpsisum from 0 to N-1gpsipsi of power seriesz457755777577979T89,899999999999ESFGkFkFSFGHI	L*ILIILL$MLJMLLLJMJM7MQQQQQQQQQQ&DT!,4l?Cdg?b@x?V=@negative size in fill_scalmatimpossible concatenation: %s %Z . %s %Zgauss_pivot_ker. k=%ld, n=%ldgauss_pivot. k=%ld, n=%ldempty matrix in supplgauss_pivot_kernegative size in fill_scalcolshallowtransgtransFpM_gauss_pivotno such component in vecextractincorrect mask in vecextractmask too large in vecextractincorrect range in extractincorrect length in sumtrying to concat elements of an empty vectorincorrect object in diagonalincorrect vector in matmuldiagonalmattodiagonalgaddmathnfdividehnf_invimageEntering gauss with inexact=%ld
Solving the triangular system
negative size in matid_FlmFpM_gauss. i=%ldnot an integer matrix in detintdetint. k=%ldempty matrix in deplinFqM_gauss_pivot* [mod p]FlxqM_kerFqM_kermissing eigenspace. Compute the matrix to higher accuracy, then restart eigen at the current precisiondet. col = %lddet, col %ld / %ldinverse mod %ld (stable=%ld)ZM_invZM_inv donegaloisindex for groups of order >127GaloisIndex: Using hash value s=%ld
Not a group in group_identGaloisIndex: Using hash value u=%ld
GaloisIndex: Using hash value w=%ld
Classification of transitive groups of order > 30 is not known-c}qeaUCWIi$	syo(7#7;89I-1_e<1QQuEF5
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	too large precision in preci()%s/galdata/%s%ld_%ld_%ldgalois files not available
[missing %s]opening %sincorrect value in bin()read_object	# rational integer roots = %ld: %ld^%ld	%2ld: %Z
indefinite invariant polynomial in gpoly()$$$$$ New prec = %ld
    ----> Group # %ld/%ld:
more than %ld rational integer roots
        all integer roots are double roots
      Working with polynomial #%ld:
tschirn
$$$$$ Tschirnhaus transformation of degree %ld: $$$$$

*** Entering isin_%ld_G_H_(%ld,%ld)
COSRES
    Output of isin_%ld_G_H(%ld,%ld): %ld
    Reordering of the roots:  )
    Output of isin_%ld_G_H(%ld,%ld): not included.
partitions( %ld ) is meaninglessPartitions of %ld (%ld)
i = %ld: %Z
%s/galdata/NAM%ldGalois names files not available, please upgrade galdata
[missing %s]galois files %s not compatible
EVENODDGaloisbig: reduced polynomial #1 = %Z
discriminant = %Z
%s group
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lsympolgaloisnamesbigbernfraccontentgchggecomprawcheckprimeidgrow_copygmillerrabinzx_copy_specrnfbasistoalgellsearchcurvesetdefaultFqX_Berlekamp_kercvtop2galoisinitgerepilecoeffs2qfb_compwr_relgtocolRgX_sqrbuchnarrowminim2factorpadic2rtodblrectlines0lllintpartialallmatmultodiagonalvecsmall_shortenfactorpadic0checksellcheck_quaddisc_realrnfpolredPerl_sv_magictruncrsprintf@@GLIBC_2.2.5ZX_resultantrnfisfreeabsi_equalisomborneconst_expressvec_isconstpolylog0padic_to_Fpexit@@GLIBC_2.2.5Perl_call_svFpM_kernumbdivqpsolublenfintfouriersinff_PHlogtraverseheapmember_zkstgalois_grouplift_intern0member_jroundrget_nffwrite@@GLIBC_2.2.5intmellininv0aprclumodiuprecreallistputgcmp1Perl_sv_setuv_mgRgX_to_FqXredrealsl2stepplothsizes_flagpari2mortalsvPerl_sv_free2gshiftispseudoprimempexp1mkvecnalgtobasisgprecprimeFpX_redbinairegsqrtgdvdvecsmall_to_vecpari_strdupPerl_push_scopegroup_isabelianidealpowsbilhellmember_tufuvecinvFqX_nbfactreducemodinvertiblesd_histsizevecteursmallresultant2isexactzeroscalarperm_to_archmatratliftprimetabzm_to_ZM_ITM_registerTMCloneTablepolhenselliftdivide_conquer_prodquotient_permrootmodINVMOD_GMP_LIMITstrtorFlv_polintform_to_idealitostrlog_base_arrayderivgpowellminimalmodelismonomecoltoalgscreen_okPL_stack_baseforeignAutoloadgen_searchisinvectoroffStackZ_chinese_coprimesommeFlxM_to_ZXMgen_0modiizcyc_powZ_factorhnfpermpolint_trivvec_shortenzeta_get_i0member_fdetach_stackinitprimesrespmPiI2nrecipRgM_to_RgXXgcvtoirootsof1Perl_newSVivgen_2u_lvalremmax_arrayRgM_to_RgXVgvarshallowtranspolzagreelouterrfactor0pari_outfilenfhilbertmulsrsqrt@@GLIBC_2.2.5PL_tmps_ixgroupelts_centervandermondeinverseprepgrealzlog_unitsisrealapproutmatlexsortstrdup@@GLIBC_2.2.5taylvecsmall_copykroneckerZ_pvalremvecgroup_sumordersmember_regispspidealmulcornacchiadiscf2CM_CardEFptype0RgXQX_divremfile_inputdo_aliasgerepileuptoFqM_supplFqM_to_FlxMouttexerrfilefamat_to_nfglengthfamat_to_archlongwordpostploth2base2hesssd_TeXstylenffromhnfbasisnewfileFqX_remis_kth_powerifac_decomp_breakZX_is_squarefreegener_Fp_localassmatzlogrnfcharpolyprint0snextprgdivmodlllkerimgennfreducemodidealpari_set_last_newlinebnfisintnormnfnewprecQ_muli_to_intFlx_POW_MONTGOMERY_LIMITpop_stackpop_val_if_newerggprecisionget_term_ftable_getmatsizesturmpartconvoldiscfgcmpsgdbllog2FpX_divremceil@@GLIBC_2.2.5gnorml2isprincipalgenforceZX_DDFRgX_divremaffirallbasesubcyclo_cyclicglcminitalg_iPerl_newSVsv_flagsqfbred0cxpsibuchrayinitgenpari_initgbittestellinit0weberf2divsrhnf_invimagesd_formatpolsubcycloFq_negPerl_newSVpvfFlxqX_safegcdterm_widthqfbrealsolveprectrmoveidealhnf0dethnf_iquadpoly0sd_prettyprintergroup_ident_transintcircnew_galois_formatsubrex01gsmithPerl_newSVpvnFlx_addintnumstepnfhilbertpFlx_SQR_LIMITPerlIO_puts__cxa_finalize@@GLIBC_2.2.5srgcd_initapprox_0Flx_INVMONTGOMERY_LIMITlegendrerectlinesFpX_to_modsd_parisizerectlinerfractoserrnfalgtobasisideallistzstarbezout_lift_factmatsolvemod0ifac_moebiusGENtostr0Flx_div_by_X_xvecpermuteimag_igp_defaultupowuuset_term_funcp2gsubstpolgzetasubgroupcondlistrnfallbaseindexpartialforellRgXQ_sqraddhelpPerl_sv_newmortalsmithrellllgramgissquareremarith_protoprodinfFpXQ_ffisom_invF2V_red_ipdiviu_remcyclognormFlxq_invmatalgtobasisFlx_powtranslate_polmathnfspecstrstr@@GLIBC_2.2.5gettimequadtocshallowconcatpowrawprimesvecteureltmul_get_tableFpV_polintpnqnFlxq_invsafeelement_mulvecpowgiFlxqX_Flxq_mulhnfmerge_get_1zncoppersmithFq_neg_invstopolyglobal_err_dataPL_markstack_maxremake_GMgerfcmuprecision0redrealsl2init_Fqrootsoldlog_gen_prcombine_factorsgrow_appendinit_graphcheckmodprfactorintbitvec_clearsqred1internsd_realprecisionisprincipalarchsv2parimatredrealnodallocatemem0ZX_caractisprimeSelfridgeFlx_subFlx_sqrspecPL_opvecsmall_lengthencbezout__ctype_b_loc@@GLIBC_2.3set_term_ftablekillallfilesforeignExprSwitchFFTinitprimitive_pol_to_monicthuehnfmodidsupplnfbasic_initrhorealquad_polmod_conjgroup_perm_normalizestderr@@GLIBC_2.2.5get_intos_getenvdivrito_famat_allquicktracesd_datadirquaddiscznstar_eltsRgX_to_RgVgcmpPerl_save_intextract_full_latticesvOutflush__sprintf_chk@@GLIBC_2.3.4monomorphismliftcopy_binpointchinvszetaKARATSUBA_MULR_LIMITZV_lincombmember_discpoldeflateQM_invgetrandFpXQ_charpolyPerl_get_hvgauss_realimagstackmallocroots0addrex01
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