bash-2.03$ ./rankdist/mwrank -b 15 -c 15 -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 May 23 2001 at 21:53:02 by GCC egcs-2.91.66 19990314 (egcs-1.1.2 release) using base arithmetic option LiDIA (LiDIA bigints and multiprecision floating point) Using LiDIA multiprecision floating point with 100 decimal places. Enter curve: [0,0,0,-24649,0] Curve [0,0,0,-24649,0] : 3 points of order 2: [157 : 0 : 1], [-157 : 0 : 1], [0 : 0 : 1] Using 2-isogenous curve [0,0,0,-271139,-54178502] ------------------------------------------------------- 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)... (-1,0,471,0,-49298): no rational point found (hlim=6) (-2,0,471,0,-24649): no rational point found (hlim=6) (157,0,471,0,314): no rational point found (hlim=6) (314,0,471,0,157): 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) = (780871468723:214710911042786809468722:53156661805) Curve E Point [-32412819426764131587825513092705845 : -167661624456834335404812111469782006 : 150201095200135518108761470235125], height = 54.60088929401703693792969682539363309246743966156908813151467356119473459202900063413388093322442211 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 = [0,0,0,-24649,0]: I. Points on E mod phi(E') Point [-8831247480342855244749962265791220 : -167661624456834335404812111469782006 : 150201095200135518108761470235125], height = 54.60088929401703693792969682539363309246743966156908813151467356119473459202900063413388093322442211 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 Height Constant = 9.5350729732492549572953066672198474407196044921875 Max height = 54.60088929401703693792969682539363309246743966156908813151467356119473459202900063413388093322442211 Bound on naive height of extra generators = 15.60183845036225906150971742559691778432709778791739868127940817346608162133655562601487565924715801 Only searching up to height 15 After point search, rank of points found is 1 Generator 1 is [-8831247480342855244749962265791220 : -167661624456834335404812111469782006 : 150201095200135518108761470235125]; height 54.60088929401703693792969682539363309246743966156908813151467356119473459202900063413388093322442211 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) = 54.60088929401703693792969682539363309246743966156908813151467356119473459202900063413388093322442211 (1745.30 seconds)