was@form:~/people/mazur/papers$ gp Reading GPRC: /etc/gprc ...Done. GP/PARI CALCULATOR Version 2.2.9 (alpha) i686 running linux (ix86 kernel) 32-bit version compiled: Apr 4 2005, gcc-3.3.5 (Debian 1:3.3.5-12) (readline v4.3 enabled, extended help available) Copyright (C) 2003 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = 4000000, primelimit = 500000 ? e=ellinit([7,13,-4,0,18]) %1 = [7, 13, -4, 0, 18, 101, -28, 88, 2026, 10873, -1151117, -22940474, -1285429208617/22940474, [-25.82509505912792226652718797, 0.2875475295639611332635939850 - 0.8770410413743752138489793091*I, 0.2875475295639611332635939850 + 0.8770410413743752138489793091*I]~, 2.141761695136367771119669349, -1.070880847568183885559834675 + 0.3073283698013708146572737348*I, -0.8932069840388415493693954490 - 7.214454502 E-28*I, 0.4466034920194207746846977246 - 1.594995609290763535086897338*I, 0.6582241302692804541155478750] ? e %2 = [7, 13, -4, 0, 18, 101, -28, 88, 2026, 10873, -1151117, -22940474, -1285429208617/22940474, [-25.82509505912792226652718797, 0.2875475295639611332635939850 - 0.8770410413743752138489793091*I, 0.2875475295639611332635939850 + 0.8770410413743752138489793091*I]~, 2.141761695136367771119669349, -1.070880847568183885559834675 + 0.3073283698013708146572737348*I, -0.8932069840388415493693954490 - 7.214454502 E-28*I, 0.4466034920194207746846977246 - 1.594995609290763535086897338*I, 0.6582241302692804541155478750] ? ?5 elladd ellak ellan ellap ellbil ellchangecurve ellchangepoint elleisnum elleta ellglobalred ellheight ellheightmatrix ellinit ellisoncurve ellj elllocalred elllseries ellminimalmodel ellorder ellordinate ellpointtoz ellpow ellrootno ellsigma ellsub elltaniyama elltors ellwp ellzeta ellztopoint ? ?elllseries elllseries(e,s,{A=1}): L-series at s of the elliptic curve e, where A a cut-off point close to 1. ? elllseries(e,1) %3 = 0.E-28 ? elllseries(e,1+0.001)/0.001 %4 = 12.84160343384561064604727440 ? ? elllseries(e,1+0.0001)/0.0001 %5 = 12.91176143829686756608037012 ? elllseries(e,1+0.000001)/0.000001 %6 = 12.91950829337230186579201596 ? ? elltors(e) %7 = [1, [], []] ? ?ellglobalred ellglobalred(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_p. ? ellglobalred(e) %8 = [22940474, [1, -8, -3, 30], 1] ? e.omega %9 = [2.141761695136367771119669349, -1.070880847568183885559834675 + 0.3073283698013708146572737348*I] ? factor(22940474) %10 = [2 1] [11470237 1] ? ================================ was@form:~$ magma Magma V2.11-10 Thu Apr 14 2005 10:18:11 on form [Seed = 186069654] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/init.m" > E = EllipticCurve([0,0,1,-1,0]) > ; Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field = Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field > E; Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field > P := E![0,0]; > 32*P; (53139223644814624290821/1870098771536627436025 : 12201668323950325956888219182513256/80871745605559864852893980186125 : 1) > Numerator((32*P)[1]); 53139223644814624290821 > Log(Numerator((32*P)[1])/Log(10)); 51.493169838232352645818628899820 > Log(Numerator((8*P)[1])/Log(10)); 2.210489992475467196697384932457 > Log(Numerator((16*P)[1])/Log(10)); 12.247729746589359868289192806687 > Log(Numerator((32*P)[1])/Log(10)); 51.493169838232352645818628899820 > Log(Numerator((64*P)[1])/Log(10)); 208.477165210801904071309619945595 > Log(Numerator((128*P)[1])/Log(10)); 836.444400683599013893158265240708 > Log(Numerator((256*P)[1])/Log(10)); 3348.57624714592440152216102602236 > 3348/836.0; 4.00478468899521531100478468899135 > 5*P; (1/4 : -5/8 : 1) > 128*P; (4220883101901813981672038828041209687388773358006621602234525283736220602784042424910652062819402630682263983416952016215705792470447590663541776879982529610225895104494227641758451306415985549983155640233811180985544569063635312166769756435837593273477418481478156083900416227929539854547219580866136447753864764557259361014879402070098960614083329079905116183621/2246298339892237851729809432418549092836609386101149882835806435061918493864250627476804879339874362689901544514304131400763535816309561054251142595975573075728979736773693621155140001012486472156148143263692074465357856616776675139517793385367445939910070342600236580812754176610494094877291148300752363483984630679270665610296274091874204309119927035346222012025 : 184957814995491979940368119688654487142220830145032444911471081323207423309754529876699063079749269488683670832370874956044699697882754707948717303519366111813601509760912561855075868920051241363507967932738683946721863693599991796208008238329228816762891187059188318742181224692148389613101169574696413147917597769027453244156901507949603191837561243939961049327979098589079120753160359964106138907333847006697847634832566522083012819664107426298030834305546887052988746082950902835702670641567311982155121575285442836765989811937091671562630581/106463601652551359596410007961826622903692200567213734808176618881015835655144344339650061204838221402766553742640815169816243674645528663038301035465329299736604170571831081074725662964906834730166904233134578710182035210976364333394779344667640367417075740951697607939581601263911504541832116590408975318043608536152301276149035149647434119212247152293224809354032750723014080137586160372264625876068024727063926662965637979869513688434642236542260926405979938462210676255598594513900131182512416510488538501523543273723137852660179435846733875 : 1) > TracesOfFrobenius(E,10); [ -2, -3, -2, -1 ] > TracesOfFrobenius(E,100); [ -2, -3, -2, -1, -5, -2, 0, 0, 2, 6, -4, -1, -9, 2, -9, 1, 8, -8, 8, 9, -1, 4, -15, 4, 4] > time v := TracesOfFrobenius(E,1000); Time: 0.000 > time v := TracesOfFrobenius(E,10000); Time: 0.080 > time v := TracesOfFrobenius(E,100000); Time: 0.890 > time v := TracesOfFrobenius(E,1000000); Time: 11.860 > EC('389a') > ; >> EC('389a') ^ User error: Identifier '389a' has not been declared or assigned > Total time: 13.890 seconds, Total memory usage: 3.56MB was@form:~$ ----- was@form:~$ mwrank Program mwrank: uses 2-descent (via 2-isogeny if possible) to determine the rank of an elliptic curve E over Q, and list a set of points which generate E(Q) modulo 2E(Q). and finally saturate to obtain generating points on the curve. For more details see the file mwrank.doc. For details of algorithms see the author's book. Please acknowledge use of this program in published work, and send problems to John.Cremona@nottingham.ac.uk. Version compiled on Feb 22 2005 at 15:18:02 by GCC 3.3.5 (Debian 1:3.3.5-8) using base arithmetic option NTL (NTL bigints, no multiprecision floating point) Enter curve: [7, 13, -4, 0, 18] Curve [7,13,-4,0,18] : Working with minimal curve [1,1,0,-226,1238] [u,r,s,t] = [1,-8,-3,30] No points of order 2 Basic pair: I=10873, J=-2302234 disc=-158564556288 2-adic index bound = 2 By Lemma 5.1(b), 2-adic index = 1 2-adic index = 1 One (I,J) pair *** BSD give two (I,J) pairs Looking for quartics with I = 10873, J = -2302234 Looking for Type 3 quartics: Trying positive a from 1 up to 36 (square a first...) (1,1,104,129,37) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [-1132:1183:64] height = 6.03222412823 Rank of B=im(eps) increases to 1 Trying positive a from 1 up to 36 (...then non-square a) Trying negative a from -1 down to -2 Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 1 Selmer rank contribution from B=im(eps) = 1 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 1 Points generating E(Q)/2E(Q): Point [-1644:6499:64], height = 6.03222412823001 Regulator (before saturation) = 6.03222412823 Saturating...finished saturation (index was 1) Regulator (after saturation) = 6.03222412823 Transferring points back to original curve [7,13,-4,0,18] Generator 1 is [-1644:6499:64]; height 6.03222412823001 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 6.03222412823 (0.022 seconds) Enter curve: bad ZZ input Aborted was@form:~$