bash-2.05$ gp Reading GPRC: /home/his/.gprc ...Done. GP/PARI CALCULATOR Version 2.1.3 (released) i386 running netbsd 32-bit version (readline v1.0 enabled, extended help available) Copyright (C) 2000 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. realprecision = 28 significant digits seriesprecision = 16 significant terms format = g0.28 parisize = 4000000, primelimit = 500000 (13:20) gp> read("theta-congr.gp") (13:21) gp> e=ec1(67) %1 = [0, 134, 0, -13467, 0, 536, -26934, 0, -181360089, 933712, -673709120, 208416112517376, 35152/9, [67.00000000000000000000000000, 0.E-28, -201.0000000000000000000000000]~, 0.2634603184007091938891808876, 0.2059471841708992395764945741*I, -5.115973846927726724687149328, -15.92351017839363847422519771*I, 0.05425891071539461015764915424] (13:21) gp> elltors(e) %2 = [4, [2, 2], [[67, 0], [0, 0]]] \\ ねじれ点群は、Z/2Z×Z/2Zであることが分かる。 \\ ここで、mwrankを使って、Mordell-Weil群のrankを求める。 --------------------- Enter curve: [0,134,0,-13467,0] Curve [0,134,0,-13467,0] : Working with minimal curve [0,-1,0,-19452,786240] [u,r,s,t] = [1,-45,0,0] 3 points of order 2: [112 : 0 : 1], [-156 : 0 : 1], [45 : 0 : 1] Using 2-isogenous curve [0,-1,0,-109232,-13219440] ------------------------------------------------------- First step, determining Selmer group ------------------------------------------------------- ------------------------------------------------------- Rank = 0 ------------------------------------------------------- Second step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q)) ------------------------------------------------------- 1. E(Q)/phi(E'(Q)) ------------------------------------------------------- This component of the rank is 0 ------------------------------------------------------- 2. E'(Q)/phi'(E(Q)) ------------------------------------------------------- This component of the rank is 0 ------------------------------------------------------- Summary of results: ------------------------------------------------------- rank(E) = 0 #E(Q)/2E(Q) = 4 Information on III(E/Q): #III(E/Q)[phi'] = 1 #III(E/Q)[2] = 1 Information on III(E'/Q): #phi'(III(E/Q)[2]) = 1 #III(E'/Q)[phi] = 1 #III(E'/Q)[2] = 1 Rank = 0 After descent, rank of points found is 0 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 1 (0.80 seconds) --------------------- \\ 位数3以上のねじれ点を持たず、Mordell-Weil群のrankは0なので、67はπ/3-非合同数である。 (13:21) gp> e=ec2(67) %3 = [0, -134, 0, -13467, 0, -536, -26934, 0, -181360089, 933712, 673709120, 208416112517376, 35152/9, [201.0000000000000000000000000, 0.E-28, -67.00000000000000000000000000]~, 0.2059471841708992395764945741, 0.2634603184007091938891808878*I, -7.925186829643808453654397871, -25.39274775795553865756622278*I, 0.05425891071539461015764915426] (13:25) gp> elltors(e) %4 = [4, [2, 2], [[201, 0], [0, 0]]] \\ ねじれ点群は、Z/2Z×Z/2Zであることが分かる。 \\ ここで、mwrankを使って、Mordell-Weil群のrankを求める。 --------------------- Enter curve: [0,-134,0,-13467,0] Curve [0,-134,0,-13467,0] : Working with minimal curve [0,1,0,-19452,-786240] [u,r,s,t] = [1,45,0,0] 3 points of order 2: [156 : 0 : 1], [-112 : 0 : 1], [-45 : 0 : 1] Using 2-isogenous curve [0,1,0,-288792,-59825568] ------------------------------------------------------- First step, determining Selmer group ------------------------------------------------------- ------------------------------------------------------- Rank <= 1 ------------------------------------------------------- Second step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q)) ------------------------------------------------------- 1. E(Q)/phi(E'(Q)) ------------------------------------------------------- First stage (no second descent yet)... (-1,0,469,0,-53868): no rational point found (hlim=6) (-3,0,469,0,-17956): no rational point found (hlim=6) (67,0,469,0,804): no rational point found (hlim=6) (201,0,469,0,268): no rational point found (hlim=6) After first descent, this component of the rank has lower bound 0 and upper bound 1 (difference = 1) Second descent will attempt to reduce this Second stage (using second descent)... d1=-1: (x:y:z) = (953611:27789263040:58361) Curve E Point [-53071972472712881 : -26500146916837440 : 198777934899881], height = 22.6400010946215 Second descent successfully found rational point for d1=-1 ------------------------------------------------------- 2. E'(Q)/phi'(E(Q)) ------------------------------------------------------- This component of the rank is 0 ------------------------------------------------------- Summary of results: ------------------------------------------------------- rank(E) = 1 #E(Q)/2E(Q) = 8 Information on III(E/Q): #III(E/Q)[phi'] = 1 #III(E/Q)[2] = 1 Information on III(E'/Q): #phi'(III(E/Q)[2]) = 1 #III(E'/Q)[phi] = 1 #III(E'/Q)[2] = 1 ------------------------------------------------------- List of points on E = [0,1,0,-19452,-786240]: I. Points on E mod phi(E') Point [-22062614628331445 : -26500146916837440 : 198777934899881], height = 22.6400010946215 II. Points on phi(E') mod 2E --none (modulo 2-torsion). ------------------------------------------------------- Computing full set of 2 coset representatives for 2E(Q) in E(Q) (modulo torsion), and sorting into height order....done. Rank = 1 After descent, rank of points found is 1 Transferring points back to original curve [0,-134,0,-13467,0] Generator 1 is [-13117607557836800 : -26500146916837440 : 198777934899881]; height 22.6400010946215 The rank has been determined unconditionally. The basis given is for a subgroup of full rank of the Mordell-Weil group (modulo torsion), possibly of index greater than 1. Regulator (of this subgroup) = 22.6400010946215 (9.10 seconds) --------------------- \\ Mordell-Weil群のrankが1なので、67は2π/3-合同数である。 (13:25) gp> g=[-13117607557836800/198777934899881, -26500146916837440/198777934899881] %5 = [-224766668800/3406006321, -26500146916837440/198777934899881] (13:27) gp> ellisoncurve(e,g) %6 = 1 \\ Mordell-Weil群の自由部分群の生成元から、2π/3-有理三角形を求める。 (13:28) gp> search2(e,g,67,6) 2Pi/3-congruent number 2([-224766668800/3406006321, -26500146916837440/198777934899881]):[6828808371/3380269120, 13521076480/101922513, 46056826394932825123/344525523326698560] 4([-224766668800/3406006321, -26500146916837440/198777934899881]):[8505115188033888352905321542070934663680/2088451379545949870481153917764259279671, 2088451379545949870481153917764259279671/31735504432962269973527319186831845760, 4502657871105567436612263016819553212988136613503370605687384523764112621106641/66278058013606660316013260547009084138992382636474469808531645829140975544960] 6([-224766668800/3406006321, -26500146916837440/198777934899881]):[91252236527750011321323517156583718826650973734248950355632202847318952762162305499514589/14729586311536924424361788785476268422685319339421138857825926387990553689529764899730240, 58918345246147697697447155141905073690741277357684555431303705551962214758119059598920960/1361973679518656885392888315769906251144044384093267915755704520109238100927795604470367, 936192156293841658387496047173961840749347321268162994269056196423154321403376604272071648893174548105029001389561366060304739451170045306410518145463474507904840690277776719363/20061308866511586462060445841523166430727179927671440916116433248951001012763199304394101436741618972489259481674781567575854298144432780649552183789671398550264384456373798080] (13:28) gp> quit; Good bye! bash-2.05$