bash-2.05a$ mwrank3 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 search for further 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 11 2003 at 17:40:15 by GCC 3.2.1 using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point) Using LiDIA multiprecision floating point with 15 decimal places. Enter curve: [0,0,0,-25921,0] Curve [0,0,0,-25921,0] : 3 points of order 2: [0 : 0 : 1], [161 : 0 : 1], [-161 : 0 : 1] **************************** * Using 2-isogeny number 1 * **************************** Using 2-isogenous curve [0,0,0,103684,0] ------------------------------------------------------- First step, determining 1st descent Selmer groups ------------------------------------------------------- After first local descent, rank bound = 2 rk(S^{phi}(E'))= 3 rk(S^{phi'}(E))= 1 ------------------------------------------------------- Second step, determining 2nd descent Selmer groups ------------------------------------------------------- After second local descent, rank bound = 2 rk(phi'(S^{2}(E)))= 3 rk(phi(S^{2}(E')))= 1 rk(S^{2}(E))= 4 rk(S^{2}(E'))= 3 **************************** * Using 2-isogeny number 2 * **************************** Using 2-isogenous curve [0,-966,0,25921,0] ------------------------------------------------------- First step, determining 1st descent Selmer groups ------------------------------------------------------- After first local descent, rank bound = 2 rk(S^{phi}(E'))= 3 rk(S^{phi'}(E))= 1 ------------------------------------------------------- Second step, determining 2nd descent Selmer groups ------------------------------------------------------- After second local descent, rank bound = 2 rk(phi'(S^{2}(E)))= 3 rk(phi(S^{2}(E')))= 1 rk(S^{2}(E))= 4 rk(S^{2}(E'))= 3 **************************** * Using 2-isogeny number 3 * **************************** Using 2-isogenous curve [0,966,0,25921,0] ------------------------------------------------------- First step, determining 1st descent Selmer groups ------------------------------------------------------- After first local descent, rank bound = 2 rk(S^{phi}(E'))= 2 rk(S^{phi'}(E))= 2 ------------------------------------------------------- Second step, determining 2nd descent Selmer groups ------------------------------------------------------- After second local descent, rank bound = 2 rk(phi'(S^{2}(E)))= 2 rk(phi(S^{2}(E')))= 2 rk(S^{2}(E))= 4 rk(S^{2}(E'))= 3 After second local descent, combined upper bound on rank = 2 Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q)) ------------------------------------------------------- 1. E(Q)/phi(E'(Q)) ------------------------------------------------------- (c,d) =(-483,51842) (c',d')=(966,25921) This component of the rank is 0 ------------------------------------------------------- 2. E'(Q)/phi'(E(Q)) ------------------------------------------------------- First stage (no second descent yet)... (-7,0,966,0,-3703): (x:y:z) = (2:7:1) Curve E' Point [-28 : -98 : 1], height = 3.12473627259575 Curve E Point [196 : 25137 : 64], height = 6.2494725451915 (23,0,966,0,1127): (x:y:z) = (1:46:1) Curve E' Point [23 : 1058 : 1], height = 1.47329413590408 Curve E Point [529 : -6348 : 1], height = 2.94658827180815 After first global descent, this component of the rank = 2 ------------------------------------------------------- Summary of results: ------------------------------------------------------- rank(E) = 2 #E(Q)/2E(Q) = 16 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 = [0,0,0,-25921,0]: I. Points on E mod phi(E') --none (modulo torsion). II. Points on phi(E') mod 2E Point [-10108 : 25137 : 64], height = 6.2494725451915 Point [368 : -6348 : 1], height = 2.94658827180815 ------------------------------------------------------- Computing full set of 4 coset representatives for 2E(Q) in E(Q) (modulo torsion), and sorting into height order....done. Rank = 2 After descent, rank of points found is 2 Generator 1 is [-10108 : 25137 : 64]; height 6.2494725451915 Generator 2 is [368 : -6348 : 1]; height 2.94658827180815 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) = 15.06262850047 (17.2 seconds) Enter curve: [0,0,0,0,0] bash-2.05a$