bash-2.05a$ 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 Apr 15 2005 at 07:36:24 by GCC 3.3.3 (NetBSD nb3 20040520) using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point) Using LiDIA multiprecision floating point with 15 decimal places. Enter curve: [1, -1, 1, -1473098895734027, 21734553366029753094651] Curve [1,-1,1,-1473098895734027,21734553366029753094651] : 1 points of order 2: [182376790 : -91188399 : 8] Using 2-isogenous curve [0,-547130364,0,69332359726080000,0] ------------------------------------------------------- First step, determining 1st descent Selmer groups ------------------------------------------------------- After first local descent, rank bound = 8 rk(S^{phi}(E'))= 5 rk(S^{phi'}(E))= 5 ------------------------------------------------------- Second step, determining 2nd descent Selmer groups ------------------------------------------------------- After second local descent, rank bound = 8 rk(phi'(S^{2}(E)))= 5 rk(phi(S^{2}(E')))= 5 rk(S^{2}(E))= 9 rk(S^{2}(E'))= 9 Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q)) ------------------------------------------------------- 1. E(Q)/phi(E'(Q)) ------------------------------------------------------- (c,d) =(273565182,1376387269153281) (c',d')=(-547130364,69332359726080000) First stage (no second descent yet)... (-231,0,273565182,0,-5958386446551): (x:y:z) = (149:34320:1) Curve E Point [-5128431 : -1181260080 : 1], height = 8.48589624739836 (-1239,0,273565182,0,-1110885608679): (x:y:z) = (68:357195:1) Curve E Point [-5729136 : -30094393140 : 1], height = 11.1305396132657 (889,0,273565182,0,1548242147529): (x:y:z) = (11:1257520:1) Curve E Point [107569 : 12297288080 : 1], height = 7.4529926117683 (489,0,273565182,0,2814697891929): (x:y:z) = (49:1863960:1) Curve E Point [1174089 : 44662345560 : 1], height = 6.56890990724921 After first global descent, this component of the rank = 5 ------------------------------------------------------- 2. E'(Q)/phi'(E(Q)) ------------------------------------------------------- First stage (no second descent yet)... (15,0,-547130364,0,4622157315072000): (x:y:z) = (110:67937760:1) Curve E' Point [181500 : 112097304000 : 1], height = 11.4880575315936 Curve E Point [95362380864 : -29490877643885640 : 1], height = 22.9761150631871 (22,0,-547130364,0,3151470896640000): (x:y:z) = (172:55993784:1) Curve E' Point [650848 : 211880478656 : 1], height = 12.2380318786475 Curve E Point [2106532271960347 : -344651762745993417176 : 79507], height = 24.3027769621551 (177,0,-547130364,0,391708247040000): (x:y:z) = (8:19790736:1) Curve E' Point [11328 : 28023682176 : 1], height = 10.9797029514561 Curve E Point [1529973560241 : -1892625608360179728 : 1], height = 21.7861191077722 (163,0,-547130364,0,425351900160000): (x:y:z) = (440:18042640:1) Curve E' Point [31556800 : 1294018140800 : 1], height = 6.70615250292232 Curve E Point [420373009 : -11099856731600 : 1], height = 13.2390182107047 After first global descent, this component of the rank = 5 ------------------------------------------------------- Summary of results: ------------------------------------------------------- rank(E) = 8 #E(Q)/2E(Q) = 512 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 = [1,-1,1,-1473098895734027,21734553366029753094651]: I. Points on E mod phi(E') Point [21514991 : -158415006 : 1], height = 8.52055360642636 Point [170918518 : -30179852403 : 8], height = 11.1305396132657 Point [22823991 : 1525749014 : 1], height = 7.48764997079629 Point [23090621 : 5571247884 : 1], height = 6.53425254822121 II. Points on phi(E') mod 2E Point [190907138518 : -29490973097454903 : 8], height = 22.9761150631871 Point [528445596920403 : -43081734566047677102 : 79507], height = 24.2681196031271 Point [382516187159 : -236578392303116046 : 1], height = 21.8207764668002 Point [127890351 : -1387546036626 : 1], height = 13.2736755697327 Rank = 8 Regulator (before saturation) = 1939034.75293001 Saturating...Warning: point search bound of 20.245 is large Saturation index bound = 222798764 WARNING: saturation at primes p > 100 will not be done; points may be unsaturated at primes between 100 and index bound Failed to saturate MW basis at primes [ ] finished saturation (index was 7) Regulator (after saturation) = 39572.1378148982 saturation possibly incomplete at primes [ ] Generator 1 is [21514991 : -158415006 : 1]; height 8.52055360642636 Generator 2 is [170918518 : -30179852403 : 8]; height 11.1305396132657 Generator 3 is [20007639 : 16436650866 : 1]; height 5.17140669154387 Generator 4 is [23090621 : 5571247884 : 1]; height 6.53425254822121 Generator 5 is [190907138518 : -29490973097454903 : 8]; height 22.9761150631871 Generator 6 is [528445596920403 : -43081734566047677102 : 79507]; height 24.2681196031271 Generator 7 is [382516187159 : -236578392303116046 : 1]; height 21.8207764668002 Generator 8 is [127890351 : -1387546036626 : 1]; height 13.2736755697327 Regulator = 39572.1378148982 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 (but not divisible by any prime less than 100 unless listed above). (113 seconds) Enter curve: [0,0,0,0,0] bash-2.05a$