$ mwrank3 -p 1000 -b 18 -c 18 -x 8 NB: reducing hlimc to 15 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 Mar 22 2003 at 13:23:23 by GCC 3.2.1 using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point) Using LiDIA multiprecision floating point with 1000 decimal places. Enter curve: [0,-4078,0,-12472563,0] Curve [0,-4078,0,-12472563,0] : Working with minimal curve [0,-1,0,-18015924,-21971882556] [u,r,s,t] = [1,1359,0,0] 3 points of order 2: [-1359 : 0 : 1], [-3398 : 0 : 1], [4758 : 0 : 1] **************************** * Using 2-isogeny number 1 * **************************** Using 2-isogenous curve [0,8156,0,66520336,0] ------------------------------------------------------- First step, determining 1st descent Selmer groups ------------------------------------------------------- After first local descent, rank bound = 1 rk(S^{phi}(E'))= 3 rk(S^{phi'}(E))= 0 ------------------------------------------------------- Second step, determining 2nd descent Selmer groups ------------------------------------------------------- After first local descent, rank bound = 1 rk(S^{phi}(E'))= 2 rk(S^{phi'}(E))= 1 ------------------------------------------------------- Second step, determining 2nd descent Selmer groups ------------------------------------------------------- After second local descent, rank bound = 1 rk(phi'(S^{2}(E)))= 2 rk(phi(S^{2}(E')))= 1 rk(S^{2}(E))= 3 rk(S^{2}(E'))= 3 **************************** * Using 2-isogeny number 3 * **************************** Using 2-isogenous curve [0,-28546,0,4157521,0] ------------------------------------------------------- First step, determining 1st descent Selmer groups ------------------------------------------------------- After first local descent, rank bound = 1 rk(S^{phi}(E'))= 3 rk(S^{phi'}(E))= 0 ------------------------------------------------------- Second step, determining 2nd descent Selmer groups ------------------------------------------------------- After second local descent, rank bound = 1 rk(phi'(S^{2}(E)))= 3 rk(phi(S^{2}(E')))= 0 rk(S^{2}(E))= 3 rk(S^{2}(E'))= 2 After second local descent, combined upper bound on rank = 1 Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q)) ------------------------------------------------------- 1. E(Q)/phi(E'(Q)) ------------------------------------------------------- (c,d) =(14273,49890252) (c',d')=(-28546,4157521) First stage (no second descent yet)... (2,0,14273,0,24945126): no rational point found (hlim=8) (6,0,14273,0,8315042): no rational point found (hlim=8) (-4078,0,14273,0,-12234): no rational point found (hlim=8) (-12234,0,14273,0,-4078): no rational point found (hlim=8) After first global 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=2: (x:y:z) = (25319035061233:922290664784376619029801461:55790283257) Curve E Point [71529116760879666645581363559036442546 : 46703019356647046694277885494323935722826 : 173650364485145856786552841575593], height = 53.94874158832206823343092244249232300920513942477712169515010215347498647488732367721653256589756729435042561214865764163413534967959577479695370373820507017087351298773152323875700136847579701999673369144784676057058297294284734942046136665972322000317900406326707518913678504113873060321324074439542588725526939063154778339091863416227528886151154031532528216715187032043628786348013846858964577846078744402086954195472835142165977703874567315605466865343579870128179429992318124918624639736248974096060291009313028298742556031256218329025713604397835665305920097345779077668978320863897037388387700029152372546228443876030489797757603440196893599781264486458953790533714963843170388076886428707685292781796123258553583676265817886477844388221824997530787200541941277951643967376799741530334198538983219692124920975938883844186732276773116473781753572245866376678797561417076348059922901844891322599018057120305041462367773221916744085289881001481460171914155599652243838752502493215385593740218112 Second descent successfully found rational point for d1=2 After second global descent, this component of the rank = 3 ------------------------------------------------------- 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,-18015924,-21971882556]: I. Points on E mod phi(E') Point [72355345195099990632171781979253114040 : 46703019356647046694277885494323935722826 : 173650364485145856786552841575593], height = 53.94874158832206823343092244249232300920513942477712169515010215347498647488732367721653256589756729435042561214865764163413534967959577479695370373820507017087351298773152323875700136847579701999673369144784676057058297294284734942046136665972322000317900406326707518913678504113873060321324074439542588725526939063154778339091863416227528886151154031532528216715187032043628786348013846858964577846078744402086954195472835142165977703874567315605466865343579870128179429992318124918624639736248974096060291009313028298742556031256218329025713604397835665305920097345779077668978320863897037388387700029152372546228443876030489797757603440196893599781264486458953790533714963843170388076886428707685292781796123258553583676265817886477844388221824997530787200541941277951643967376799741530334198538983219692124920975938883844186732276773116473781753572245866376678797561417076348059922901844891322599018057120305041462367773221916744085289881001481460171914155599652243838752502493215385593740218112 II. Points on phi(E') mod 2E --none (modulo 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 = 9.035954890608440592814076808281242847442626953125 Max height = 53.94874158832206823343092244249232300920513942477712169515010215347498647488732367721653256589756729435042561214865764163413534967959577479695370373820507017087351298773152323875700136847579701999673369144784676057058297294284734942046136665972322000317900406326707518913678504113873060321324074439542588725526939063154778339091863416227528886151154031532528216715187032043628786348013846858964577846078744402086954195472835142165977703874567315605466865343579870128179429992318124918624639736248974096060291009313028298742556031256218329025713604397835665305920097345779077668978320863897037388387700029152372546228443876030489797757603440196893599781264486458953790533714963843170388076886428707685292781796123258553583676265817886477844388221824997530787200541941277951643967376799741530334198538983219692124920975938883844186732276773116473781753572245866376678797561417076348059922901844891322599018057120305041462367773221916744085289881001481460171914155599652243838752502493215385593740218112 Bound on naive height of extra generators = 15.03025951153311484097306819078038984846542022254468018835001135038610960832081374191294806287750747715004729023873973795934837218662175275521707819313389668565261255419239147097300015205286633555519263238309408450784255254920526104671792962885813555590877822925189724323742056012652562257924897159949176525058548784794975371010207046247503209572350447948058690746131892449292087372001538539884953094008749378009661577274759460240664189319396368400607429482619985569797714443590902768736071081805441566228921223257003144304728447917357592113968178266426185033991121927308786407664257873766337487598633336572485838469827097336721088639733715577432622197918276273217087837079440427018932008542936523076143642421791473172620408473979765164204932024647221947865244504660141994627107486311082392259355393220357743569435663993209316020748030752568497086861508027318486297644173490786260895546989093876813622113117457811671273596419246879638231698875666831273352434906177739138204305833610357265065971135346 After point search, rank of points found is 1 Transferring points back to original curve [0,-4078,0,-12472563,0] Generator 1 is [72591336040435303851544707290954344927 : 46703019356647046694277885494323935722826 : 173650364485145856786552841575593]; height 53.94874158832206823343092244249232300920513942477712169515010215347498647488732367721653256589756729435042561214865764163413534967959577479695370373820507017087351298773152323875700136847579701999673369144784676057058297294284734942046136665972322000317900406326707518913678504113873060321324074439542588725526939063154778339091863416227528886151154031532528216715187032043628786348013846858964577846078744402086954195472835142165977703874567315605466865343579870128179429992318124918624639736248974096060291009313028298742556031256218329025713604397835665305920097345779077668978320863897037388387700029152372546228443876030489797757603440196893599781264486458953790533714963843170388076886428707685292781796123258553583676265817886477844388221824997530787200541941277951643967376799741530334198538983219692124920975938883844186732276773116473781753572245866376678797561417076348059922901844891322599018057120305041462367773221916744085289881001481460171914155599652243838752502493215385593740218112 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 53.94874158832206823343092244249232300920513942477712169515010215347498647488732367721653256589756729435042561214865764163413534967959577479695370373820507017087351298773152323875700136847579701999673369144784676057058297294284734942046136665972322000317900406326707518913678504113873060321324074439542588725526939063154778339091863416227528886151154031532528216715187032043628786348013846858964577846078744402086954195472835142165977703874567315605466865343579870128179429992318124918624639736248974096060291009313028298742556031256218329025713604397835665305920097345779077668978320863897037388387700029152372546228443876030489797757603440196893599781264486458953790533714963843170388076886428707685292781796123258553583676265817886477844388221824997530787200541941277951643967376799741530334198538983219692124920975938883844186732276773116473781753572245866376678797561417076348059922901844891322599018057120305041462367773221916744085289881001481460171914155599652243838752502493215385593740218112 (639 seconds)