bash-2.05a$ mwrank3 -b 15 -c 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 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, 12, 0, 0, -1728] Curve [0,12,0,0,-1728] : Working with minimal curve [0,0,0,-3,-25] [u,r,s,t] = [2,-4,0,0] No points of order 2 Basic pair: I=9, J=675 disc=-452709 2-adic index bound = 2 2-adic index = 2 Two (I,J) pairs Looking for quartics with I = 9, J = 675 Looking for Type 3 quartics: Trying positive a from 1 up to 1 (square a first...) Trying positive a from 1 up to 1 (...then non-square a) Trying negative a from -1 down to -2 Finished looking for Type 3 quartics. Looking for quartics with I = 144, J = 43200 Looking for Type 3 quartics: Trying positive a from 1 up to 5 (square a first...) Trying positive a from 1 up to 5 (...then non-square a) (3,0,-24,24,-12) --nontrivial...not locally soluble (p = 2) Trying negative a from -1 down to -10 (-1,0,0,40,-12) --nontrivial...not locally soluble (p = 2) (-3,0,-6,24,-3) --nontrivial...not locally soluble (p = 2) (-3,-4,48,144,108) --nontrivial...not locally soluble (p = 2) (-9,-12,-6,12,3) --nontrivial...not locally soluble (p = 2) Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 0 Selmer rank contribution from B=im(eps) = 0 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 0 After descent, rank of points found is 0 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 1 (3.1 seconds) Enter curve: [0, 12, 0, 0, -3888] Curve [0,12,0,0,-3888] : Working with minimal curve [0,0,1,-3,-59] [u,r,s,t] = [2,-4,0,4] No points of order 2 Basic pair: I=144, J=101520 disc=-10294366464 2-adic index bound = 2 By Lemma 5.1(a), 2-adic index = 1 2-adic index = 1 One (I,J) pair Looking for quartics with I = 144, J = 101520 Looking for Type 3 quartics: Trying positive a from 1 up to 8 (square a first...) Trying positive a from 1 up to 8 (...then non-square a) (2,2,-60,144,-108) --trivial (3,6,-24,0,-12) --nontrivial...not locally soluble (p = 3) (5,4,-24,-4,-8) --trivial (6,-6,-24,24,-12) --nontrivial...not locally soluble (p = 3) Trying negative a from -1 down to -13 (-2,-2,-24,8,20) --trivial (-3,0,-24,12,12) --nontrivial...not locally soluble (p = 3) (-3,0,-6,36,-3) --nontrivial...not locally soluble (p = 3) (-3,0,12,36,0) --trivial (-3,0,30,60,21) --nontrivial...not locally soluble (p = 3) (-3,0,48,108,60) --nontrivial...not locally soluble (p = 3) (-3,0,102,324,285) --nontrivial...not locally soluble (p = 3) (-3,3,3,33,-12) --nontrivial...not locally soluble (p = 3) (-3,4,-24,36,0) --trivial (-3,-4,0,36,8) --trivial (-3,5,72,144,80) --trivial (-3,6,-24,48,-12) --nontrivial...not locally soluble (p = 3) (-3,6,12,24,-12) --nontrivial...not locally soluble (p = 3) (-5,1,-15,27,0) --trivial Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 0 Selmer rank contribution from B=im(eps) = 0 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 0 After descent, rank of points found is 0 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 1 (4 seconds) Enter curve: [0, 12, 0, 0, -6912] Curve [0,12,0,0,-6912] : Working with minimal curve [0,0,0,-3,-106] [u,r,s,t] = [2,-4,0,0] No points of order 2 Basic pair: I=9, J=2862 disc=-8188128 2-adic index bound = 2 After 2-adic refinement (case 2); 2-adic index = 2 2-adic index = 2 Two (I,J) pairs Looking for quartics with I = 9, J = 2862 Looking for Type 3 quartics: Trying positive a from 1 up to 2 (square a first...) Trying positive a from 1 up to 2 (...then non-square a) Trying negative a from -1 down to -4 (-3,0,3,6,0) --trivial Finished looking for Type 3 quartics. Looking for quartics with I = 144, J = 183168 Looking for Type 3 quartics: Trying positive a from 1 up to 10 (square a first...) (1,0,-30,16,-63) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [5 : 2 : 1] height = 1.43969830063111 Doubling global 2-adic index to 2 global 2-adic index is equal to local index so we abort the search for large quartics Rank of B=im(eps) increases to 1 Exiting search for large quartics after finding enough globally soluble ones. Mordell rank contribution from B=im(eps) = 1 Selmer rank contribution from B=im(eps) = 1 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 1 Points generating E(Q)/2E(Q): Point [16 : 16 : 1], height = 1.43969830063111 Height Constant = 1.00749633298967 Max height = 1.43969830063111 Bound on naive height of extra generators = 1.16746281083757 After point search, rank of points found is 1 Transferring points back to original curve [0,12,0,0,-6912] Generator 1 is [16 : 16 : 1]; height 1.43969830063111 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 1.43969830063111 (6.5 seconds) Enter curve: [0, 12, 0, 0, -10800] Curve [0,12,0,0,-10800] : Working with minimal curve [0,0,1,-3,-167] [u,r,s,t] = [2,-4,0,4] No points of order 2 Basic pair: I=144, J=288144 disc=-83015020800 2-adic index bound = 2 By Lemma 5.1(a), 2-adic index = 1 2-adic index = 1 One (I,J) pair Looking for quartics with I = 144, J = 288144 Looking for Type 3 quartics: Trying positive a from 1 up to 12 (square a first...) Trying positive a from 1 up to 12 (...then non-square a) (3,3,-33,-15,-30) --nontrivial...not locally soluble (p = 5) (5,0,-108,268,-192) --nontrivial...not locally soluble (p = 5) Trying negative a from -1 down to -19 (-2,-2,12,80,20) --nontrivial...not locally soluble (p = 5) (-3,0,-6,60,-3) --nontrivial...not locally soluble (p = 5) (-3,0,12,60,0) --trivial Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 0 Selmer rank contribution from B=im(eps) = 0 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 0 After descent, rank of points found is 0 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 1 (2.9 seconds) Enter curve: [0, 12, 0, 0, -15552] Curve [0,12,0,0,-15552] : Working with minimal curve [0,0,0,-3,-241] [u,r,s,t] = [2,-4,0,0] No points of order 2 Basic pair: I=9, J=6507 disc=-42338133 2-adic index bound = 2 2-adic index = 2 Two (I,J) pairs Looking for quartics with I = 9, J = 6507 Looking for Type 3 quartics: Trying positive a from 1 up to 3 (square a first...) (1,0,-15,27,-18) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [10 : 27 : 1] height = 1.01695473403467 Rank of B=im(eps) increases to 1 (1,-1,-15,36,-27) --trivial Trying positive a from 1 up to 3 (...then non-square a) Trying negative a from -1 down to -5 (-1,1,-6,17,-2) --trivial (-3,0,3,9,0) --trivial (-3,1,-6,9,0) --trivial Finished looking for Type 3 quartics. Looking for quartics with I = 144, J = 416448 Looking for Type 3 quartics: Trying positive a from 1 up to 14 (square a first...) (1,0,-762,11448,-48375) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [127 : 1431 : 1] height = 2.88003547365512 Doubling global 2-adic index to 2 global 2-adic index is equal to local index so we abort the search for large quartics Rank of B=im(eps) increases to 2 Exiting search for large quartics after finding enough globally soluble ones. Mordell rank contribution from B=im(eps) = 2 Selmer rank contribution from B=im(eps) = 2 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 2 Points generating E(Q)/2E(Q): Point [36 : 216 : 1], height = 1.01695473403467 Point [504 : 11448 : 1], height = 2.88003547365512 Height Constant = 2.38512483730041 Max height = 2.88003547365512 Bound on naive height of extra generators = 5.26516031095553 After point search, rank of points found is 2 Transferring points back to original curve [0,12,0,0,-15552] Generator 1 is [36 : 216 : 1]; height 1.01695473403467 Generator 2 is [504 : 11448 : 1]; height 2.88003547365512 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 0.826664218109276 (17 seconds) Enter curve: [0, 12, 0, 0, -27648] Curve [0,12,0,0,-27648] : Working with minimal curve [0,0,0,-3,-430] [u,r,s,t] = [2,-4,0,0] No points of order 2 Basic pair: I=9, J=11610 disc=-134789184 2-adic index bound = 2 After 2-adic refinement (case 2); 2-adic index = 2 2-adic index = 2 Two (I,J) pairs Looking for quartics with I = 9, J = 11610 Looking for Type 3 quartics: Trying positive a from 1 up to 4 (square a first...) Trying positive a from 1 up to 4 (...then non-square a) (3,-3,-15,24,-12) --trivial (3,6,-15,6,-3) --nontrivial...not locally soluble (p = 2) Trying negative a from -1 down to -6 (-3,0,3,12,0) --trivial (-3,-2,9,18,5) --nontrivial...not locally soluble (p = 2) (-3,-3,-6,9,3) --trivial (-3,6,-15,18,-3) --nontrivial...not locally soluble (p = 2) (-5,6,-3,10,-3) --nontrivial...not locally soluble (p = 2) Finished looking for Type 3 quartics. Looking for quartics with I = 144, J = 743040 Looking for Type 3 quartics: Trying positive a from 1 up to 17 (square a first...) Trying positive a from 1 up to 17 (...then non-square a) (13,12,-48,8,-12) --nontrivial...not locally soluble (p = 2) Trying negative a from -1 down to -26 (-3,0,-6,96,-3) --nontrivial...not locally soluble (p = 2) (-3,0,66,192,117) --nontrivial...not locally soluble (p = 2) (-3,0,210,960,1221) --nontrivial...not locally soluble (p = 2) (-3,-4,-48,-24,52) --nontrivial...not locally soluble (p = 2) (-3,-4,24,120,52) --nontrivial...not locally soluble (p = 2) Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 0 Selmer rank contribution from B=im(eps) = 0 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 0 After descent, rank of points found is 0 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 1 (2.8 seconds) Enter curve: [0, 12, 0, 0, -34992] Curve [0,12,0,0,-34992] : Working with minimal curve [0,0,1,-3,-545] [u,r,s,t] = [2,-4,0,4] No points of order 2 Basic pair: I=144, J=941328 disc=-886086459648 2-adic index bound = 2 By Lemma 5.1(a), 2-adic index = 1 2-adic index = 1 One (I,J) pair Looking for quartics with I = 144, J = 941328 Looking for Type 3 quartics: Trying positive a from 1 up to 18 (square a first...) (1,2,-288,2384,-5708) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [386 : 2669 : 8] height = 3.16929511350014 Rank of B=im(eps) increases to 1 Trying positive a from 1 up to 18 (...then non-square a) (5,-8,-66,172,-139) --nontrivial...not locally soluble (p = 3) (6,-10,-60,144,-108) --trivial (13,-12,-48,44,-24) --trivial Trying negative a from -1 down to -28 (-1,-1,552,7344,27216) --trivial (-1,2,-48,128,116) --nontrivial...not locally soluble (p = 3) (-3,0,12,108,0) --trivial (-6,2,-18,78,-4) --trivial (-7,-9,-54,-28,24) --trivial (-9,18,-60,78,-7) --nontrivial...not locally soluble (p = 3) (-11,-12,24,76,24) --trivial Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 1 Selmer rank contribution from B=im(eps) = 1 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 1 Points generating E(Q)/2E(Q): Point [189 : 2673 : 1], height = 3.16929511350014 Height Constant = 2.66669255711693 Max height = 3.16929511350014 Bound on naive height of extra generators = 3.01883645861694 After point search, rank of points found is 1 Transferring points back to original curve [0,12,0,0,-34992] Generator 1 is [189 : 2673 : 1]; height 3.16929511350014 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 3.16929511350014 (7.3 seconds) Enter curve: [0, 12, 0, 0, -43200] Curve [0,12,0,0,-43200] : Working with minimal curve [0,0,0,-3,-673] [u,r,s,t] = [2,-4,0,0] No points of order 2 Basic pair: I=9, J=18171 disc=-330182325 2-adic index bound = 2 2-adic index = 2 Two (I,J) pairs Looking for quartics with I = 9, J = 18171 Looking for Type 3 quartics: Trying positive a from 1 up to 4 (square a first...) Trying positive a from 1 up to 4 (...then non-square a) Trying negative a from -1 down to -7 (-3,0,3,15,0) --trivial Finished looking for Type 3 quartics. Looking for quartics with I = 144, J = 1162944 Looking for Type 3 quartics: Trying positive a from 1 up to 19 (square a first...) (1,0,-66,200,-351) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [11 : 25 : 1] height = 1.69822263116049 Doubling global 2-adic index to 2 global 2-adic index is equal to local index so we abort the search for large quartics Rank of B=im(eps) increases to 1 Exiting search for large quartics after finding enough globally soluble ones. Mordell rank contribution from B=im(eps) = 1 Selmer rank contribution from B=im(eps) = 1 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 1 Points generating E(Q)/2E(Q): Point [40 : 200 : 1], height = 1.69822263116049 Height Constant = 1.64163794491894 Max height = 1.69822263116049 Bound on naive height of extra generators = 1.83032934838122 After point search, rank of points found is 1 Transferring points back to original curve [0,12,0,0,-43200] Generator 1 is [40 : 200 : 1]; height 1.69822263116049 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 1.69822263116049 (5.1 seconds)