bash-2.05a$ mwrank3 -p 100 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 100 decimal places. Enter curve: [0,0,0,-173027716,0] Curve [0,0,0,-173027716,0] : 3 points of order 2: [0 : 0 : 1], [13154 : 0 : 1], [-13154 : 0 : 1] **************************** * Using 2-isogeny number 1 * **************************** Using 2-isogenous curve [0,0,0,692110864,0] ------------------------------------------------------- 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,-78924,0,173027716,0] ------------------------------------------------------- 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 [0,78924,0,173027716,0] ------------------------------------------------------- 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) =(-39462,346055432) (c',d')=(78924,173027716) First stage (no second descent yet)... (6577,0,-39462,0,52616): (x:y:z) = (2:0:1) Curve E Point [26308 : 0 : 1], height = 0 After first global descent, this component of the rank = 2 ------------------------------------------------------- 2. E'(Q)/phi'(E(Q)) ------------------------------------------------------- First stage (no second descent yet)... (2,0,78924,0,86513858): no rational point found (hlim=8) (6577,0,78924,0,26308): no rational point found (hlim=8) (13154,0,78924,0,13154): (x:y:z) = (7:26308:11) Curve E' Point [7090006 : 2422388024 : 1331], height = 4.417409513260499328697212338234745933335553872785258752604756178881220907389844692018151204278228186 Curve E Point [13323134132 : -529464810960 : 456533], height = 8.834819026520998657394424676469491866671107745570517505209512357762441814779689384036302408556456373 After first global descent, this component of the rank has lower bound 1 and upper bound 2 (difference = 1) Second descent will attempt to reduce this Second stage (using second descent)... d1=2: Second descent inconclusive for d1=2: ELS descendents exist but no rational point found d1=6577: (x:y:z) = (6341276706:3384599886226294689442:507945161) Curve E' Point [134337752401400507314987365589092 : 141160075414930435994956836320168004 : 131054060667896545406208281], height = 44.47152468273849592002491523219023579384969119815097282575265065404384762793429163095898649903621351 Curve E Point [9224614191365437221043635959369413842530289207203320985861106 : 4498632180352515969289398016839862069413822463336601999059827811 : 33418010796045858425237248164697246575812088605955700296], height = 88.94304936547699184004983046438047158769938239630194565150530130808769525586858326191797299807242702 Second descent successfully found rational point for d1=6577 After second 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,-173027716,0]: I. Points on E mod phi(E') --none (modulo torsion). II. Points on phi(E') mod 2E Point [7317899050 : -529464810960 : 456533], height = 8.834819026520998657394424676469491866671107745570517505209512357762441814779689384036302408556456373 Point [8785033677354249999318065197010986261072056993680579704167522 : 4498632180352515969289398016839862069413822463336601999059827811 : 33418010796045858425237248164697246575812088605955700296], height = 88.94304936547699184004983046438047158769938239630194565150530130808769525586858326191797299807242702 ------------------------------------------------------- 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 [7317899050 : -529464810960 : 456533]; height 8.834819026520998657394424676469491866671107745570517505209512357762441814779689384036302408556456373 Generator 2 is [8785033677354249999318065197010986261072056993680579704167522 : 4498632180352515969289398016839862069413822463336601999059827811 : 33418010796045858425237248164697246575812088605955700296]; height 88.94304936547699184004983046438047158769938239630194565150530130808769525586858326191797299807242702 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) = 782.2767514989986958777684231979457757176837682705658954166290660292447944443318724459633128103795557 (121 seconds) Enter curve: [0,0,0,0,0] bash-2.05a$