Curve [0,1,1,-2,0] : Basic pair: I=112, J=-1712 disc=2688768 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 = 112, J = -1712 Looking for Type 2 quartics: Trying positive a from 1 up to 2 (square a first...) (1,0,-8,4,4) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [1:0:1] height = 0.476711659343739 Rank of B=im(eps) increases to 1 Trying positive a from 1 up to 2 (...then non-square a) Trying negative a from -1 down to -1 Finished looking for Type 2 quartics. Looking for Type 1 quartics: Trying positive a from 1 up to 3 (square a first...) (1,0,4,12,8) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [-1:1:1] height = 0.686667083305586 Rank of B=im(eps) increases to 2 (The previous point is on the egg) Exiting search for Type 1 quartics after finding one which is globally soluble. 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 Used full 2-descent via multiplication-by-2 map Rank = 2 Rank of S^2(E) = 2 Searching for points (bound = 8)...done: found points which generate a subgroup of rank 2 and regulator 0.152460177943144 Processing points found during 2-descent...done: now regulator = 0.152460177943144 Saturating (with bound = 1000)...done: points were already saturated. Transferring points from minimal curve [0,1,1,-2,0] back to original curve [0,1,1,-2,0] Generator 1 is [0:-1:1]; height 0.327000773651605 Generator 2 is [-1:1:1]; height 0.686667083305586 Regulator = 0.152460177943144 The rank and full Mordell-Weil basis have been determined unconditionally. Curve [0,0,0,0,-675/4] : integral model = [0,0,0,0,-10800] with scale factor 2 Working with minimal curve [0,0,1,0,-169] via [u,r,s,t] = [2,0,0,4] Basic pair: I=0, J=291600 disc=-85030560000 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 = 0, J = 291600 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) (2,-2,-78,298,-328) --trivial Trying negative a from -1 down to -19 (-3,0,0,60,0) --trivial (-7,-4,12,44,8) --trivial (-7,12,36,36,0) --trivial (-7,-13,-42,-28,8) --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 Used full 2-descent via multiplication-by-2 map Rank = 0 Rank of S^2(E) = 0 Processing points found during 2-descent...done: now regulator = 1 Regulator = 1 The rank and full Mordell-Weil basis have been determined unconditionally. Curve [0,0,0,0,73/64] : integral model = [0,0,0,0,73] with scale factor 2 Basic pair: I=0, J=-1971 disc=-3884841 2-adic index bound = 2 2-adic index = 2 Two (I,J) pairs Looking for quartics with I = 0, J = -1971 Looking for Type 3 quartics: Trying positive a from 1 up to 3 (square a first...) (1,0,6,3,-3) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [-4:3:1] height = 0.962573188092379 Rank of B=im(eps) increases to 1 Trying positive a from 1 up to 3 (...then non-square a) Trying negative a from -1 down to -2 Finished looking for Type 3 quartics. Looking for quartics with I = 0, J = -126144 Looking for Type 3 quartics: Trying positive a from 1 up to 14 (square a first...) (1,0,-12,72,-12) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [2:9:1] height = 0.923943171600397 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 Used full 2-descent via multiplication-by-2 map Rank = 2 Rank of S^2(E) = 2 Searching for points (bound = 8)...done: found points which generate a subgroup of rank 2 and regulator 0.886543288687716 Processing points found during 2-descent...done: now regulator = 0.886543288687716 Saturating (with bound = 1000)...done: points were already saturated. Transferring points from minimal curve [0,0,0,0,73] back to original curve [0,0,0,0,73/64] Generator 1 is [-8:3:8]; height 0.962573188092379 Generator 2 is [4:9:8]; height 0.923943171600397 Regulator = 0.886543288687716 The rank and full Mordell-Weil basis have been determined unconditionally. Curve [0,0,0,677,0] : 1 points of order 2: [0:0:1] Using 2-isogenous curve [0,0,0,-2708,0] ------------------------------------------------------- First step, determining 1st descent Selmer groups ------------------------------------------------------- After first local descent, rank bound = 1 rk(S^{phi}(E'))= 1 rk(S^{phi'}(E))= 2 ------------------------------------------------------- Second step, determining 2nd descent Selmer groups ------------------------------------------------------- After second local descent, rank bound = 1 rk(phi'(S^{2}(E)))= 1 rk(phi(S^{2}(E')))= 2 rk(S^{2}(E))= 2 rk(S^{2}(E'))= 2 Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q)) ------------------------------------------------------- 1. E(Q)/phi(E'(Q)) ------------------------------------------------------- (c,d) =(0,677) (c',d')=(0,-2708) This component of the rank is 0 ------------------------------------------------------- 2. E'(Q)/phi'(E(Q)) ------------------------------------------------------- First stage (no second descent yet)... (-1,0,0,0,2708): (x:y:z) = (502:279022:85) Curve E' Point [-21420340:-140069044:614125], height = 12.3948826572104 Curve E Point [830499826893070:-7145241014093919:77690502163000], height = 24.7897653144207 After first global descent, this component of the rank = 2 ------------------------------------------------------- Summary of results: ------------------------------------------------------- rank(E) = 1 #E(Q)/2E(Q) = 4 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 = [0,0,0,677,0]: I. Points on E mod phi(E') --none (modulo torsion). II. Points on phi(E') mod 2E Point [830499826893070:-7145241014093919:77690502163000], height = 24.7897653144207 ------------------------------------------------------- Computing full set of 2 coset representatives for 2E(Q) in E(Q) (modulo torsion), and sorting into height order....done. Used descent via 2-isogeny with isogenous curve E' = [0,0,0,-2708,0] Rank = 1 Rank of S^2(E) = 2 Rank of S^2(E') = 2 Rank of S^phi(E') = 1 Rank of S^phi'(E) = 2 Searching for points (bound = 8)...done: found points which generate a subgroup of rank 0 and regulator 1 Processing points found during 2-descent...done: 2-descent increases rank to 1, now regulator = 24.7897653144207 Saturating (with bound = 1000)...done: points were already saturated. Transferring points from minimal curve [0,0,0,677,0] back to original curve [0,0,0,677,0] Generator 1 is [830499826893070:-7145241014093919:77690502163000]; height 24.7897653144207 Regulator = 24.7897653144207 The rank and full Mordell-Weil basis have been determined unconditionally. Curve [0,0,0,-169,0] : 3 points of order 2: [-13:0:1], [0:0:1], [13:0:1] **************************** * Using 2-isogeny number 1 * **************************** Using 2-isogenous curve [0,78,0,169,0] (minimal model [0,0,0,-1859,30758]) ------------------------------------------------------- First step, determining 1st 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'))= 2 **************************** * Using 2-isogeny number 2 * **************************** Using 2-isogenous curve [0,0,0,676,0] ------------------------------------------------------- First step, determining 1st 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'))= 2 **************************** * Using 2-isogeny number 3 * **************************** Using 2-isogenous curve [0,-78,0,169,0] (minimal model [0,0,0,-1859,-30758]) ------------------------------------------------------- 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) =(39,338) (c',d')=(-78,169) First stage (no second descent yet)... (-1,0,39,0,-338): (x:y:z) = (19:102:5) Curve E Point [-1805:-1938:125], height = 5.83251173698765 After first 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 = [0,0,0,-169,0]: I. Points on E mod phi(E') Point [-180:-1938:125], height = 5.83251173698765 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. Used descent via 2-isogeny with isogenous curve E' = [0,0,0,-1859,-30758] Rank = 1 Rank of S^2(E) = 3 Rank of S^2(E') = 2 Rank of S^phi(E') = 3 Rank of S^phi'(E) = 0 Searching for points (bound = 8)...done: found points which generate a subgroup of rank 1 and regulator 5.83251173698765 Processing points found during 2-descent...done: now regulator = 5.83251173698765 Saturating (with bound = 1000)...done: points were already saturated. Transferring points from minimal curve [0,0,0,-169,0] back to original curve [0,0,0,-169,0] Generator 1 is [-180:1938:125]; height 5.83251173698765 Regulator = 5.83251173698765 The rank and full Mordell-Weil basis have been determined unconditionally.