was@form:~$ sage ------------------------------------------------------------- ----------- SAGE Version 0.7.11, Export Date: 2005-10-12-1029 Distributed under the terms of the GNU General Public Licen se (GPL) IPython shell -- for help type ?, ??, %magi c, or help ------------------------------------------------------------- ----------- sage: E = EllipticCurve([0,-1]) sage: E _2 = Elliptic Curve defined by y^2 = x^3 -1 over Rational F ield sage: E.gens(verbose=True) _3 = [] sage: E.gens? sage: E.gens(verbose=True, only_use_mwrank=True) _5 = [] sage: E = EllipticCurve([0,-1]) sage: E.gens(verbose=True, only_use_mwrank=True) 1 points of order 2: [1:0:1] Using 2-isogenous curve [0,-6,0,-3,0] ------------------------------------------------------- First step, determining 1st descent Selmer groups ------------------------------------------------------- After first local descent, rank bound = 0 rk(S^{phi}(E'))= 1 rk(S^{phi'}(E))= 1 ------------------------------------------------------- Second step, determining 2nd descent Selmer groups ------------------------------------------------------- ...skipping since we already know rank=0 After second local descent, rank bound = 0 rk(phi'(S^{2}(E)))= 1 rk(phi(S^{2}(E')))= 1 rk(S^{2}(E))= 1 rk(S^{2}(E'))= 1 Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q)) ------------------------------------------------------- 1. E(Q)/phi(E'(Q)) ------------------------------------------------------- (c,d) =(3,3) (c',d')=(-6,-3) This component of the rank is 0 ------------------------------------------------------- 2. E'(Q)/phi'(E(Q)) ------------------------------------------------------- This component of the rank is 0 ------------------------------------------------------- Summary of results: ------------------------------------------------------- rank(E) = 0 #E(Q)/2E(Q) = 2 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 Regulator (before saturation) = 1 Searching for points (bound = 10)...done Regulator (after searching) = 1 Saturating (bound = -1)...finished saturation (index was 0) Regulator (after saturation) = 1 _7 = [] sage: E = EllipticCurve([0,1,0,-1,0]) sage: E _9 = Elliptic Curve defined by y^2 = x^3 + x^2 - x over Rat ional Field sage: E = EllipticCurve([0,0,1,-1,0]) sage: E _11 = Elliptic Curve defined by y^2 + y = x^3 - x over Ration al Field sage: E.gens(verbose=True, only_use_mwrank=True) Basic pair: I=48, J=-432 disc=255744 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 = 48, J = -432 Looking for Type 2 quartics: Trying positive a from 1 up to 1 (square a first...) (1,0,-6,4,1) --trivial Trying positive a from 1 up to 1 (...then non-square a) Finished looking for Type 2 quartics. Looking for Type 1 quartics: Trying positive a from 1 up to 2 (square a first...) (1,0,0,4,4) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [0:0:1] height = 0.051111408239968840235886099756942021609538 202280854 Rank of B=im(eps) increases to 1 (The previous point is on th e egg) Exiting search for Type 1 quartics after finding one which is globally soluble. 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 Regulator (before saturation) = 0.051111408239968840235886099 756942021609538202280854 Searching for points (bound = 10)...done Regulator (after searching) = 0.05111140823996884023588609975 6942021609538202280854 Saturating (bound = -1)...finished saturation (index was 1) Regulator (after saturation) = 0.0511114082399688402358860997 56942021609538202280854 _12 = [(0, 0)] sage: E = EllipticCurve([7,4000000]) sage: E.gens(verbose=True, only_use_mwrank=True) Basic pair: I=-21, J=-108000000 disc=-11664000000037044 2-adic index bound = 4 2-adic index = 4 Two (I,J) pairs Looking for quartics with I = -21, J = -108000000 Looking for Type 3 quartics: Trying positive a from 1 up to 137 (square a first...) Trying positive a from 1 up to 137 (...then non-square a) (20,-1,-309,806,-408) --nontrivial...locally soluble...no r ational point found (limit 8) --new (B) #1 (70,99,165,318,80) --nontrivial...--equivalent to (B) #1 Trying negative a from -1 down to -91 Finished looking for Type 3 quartics. Looking for quartics with I = -336, J = -6912000000 Looking for Type 3 quartics: Trying positive a from 1 up to 549 (square a first...) (1,0,0,16000,-28) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [0:2000:1] height = 5.529366426734982863340020597597267739712206 3373968 Rank of B=im(eps) increases to 1 Trying positive a from 1 up to 549 (...then non-square a) (50,24,636,2056,-428) --nontrivial...not locally soluble (p = 2) (77,132,-1236,2384,-632) --nontrivial...not locally so luble (p = 2) (370,-736,1392,-560,-158) --nontrivial...not locally so luble (p = 2) Trying negative a from -1 down to -366 (-107,172,4236,11504,9352) --nontrivial...not locally so luble (p = 2) (-158,-72,2124,4568,2900) --nontrivial...not locally so luble (p = 2) Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 1 Selmer rank contribution from B=im(eps) = 2 Sha rank contribution from B=im(eps) = 1 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Warning: Selmer rank = 2 and program finds lower bound for rank = 1 which differs by an odd integer from the Selmer rank. Hence the rank must be 1 more than reported here. Try rerunning with a higher bound for quartic point search. Summary of results (all should be powers of 2): n0 = #E(Q)[2] = 1 n1 = #E(Q)/2E(Q) >= 2 n2 = #S^(2)(E/Q) = 4 #III(E/Q)[2] <= 2 1 <= rank <= selmer-rank = 2 Regulator (before saturation) = 5.529366426734982863340020597 5972677397122063373968 Searching for points (bound = 10)...done Regulator (after searching) = 5.52936642673498286334002059759 72677397122063373968 Saturating (bound = -1)...----------------------------------- ------------------------- Traceback (most recent call last): File "", line 1, in ? File "/home/was/sage/local/lib/python2.4/site-packages/sage /schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie ld.py", line 596, in gens raise RuntimeError, "Unable to compute the rank, hence ge nerators, with certainty (lower bound=%s). This could be bec ause Sha[2] is nontrivial."%C.rank() RuntimeError: Unable to compute the rank, hence generators, w ith certainty (lower bound=1). This could be because Sha[2] is nontrivial. finished saturation (index was 1) Regulator (after saturation) = 5.5293664267349828633400205975 972677397122063373968 sage: E.gens() ------------------------------------------------------------ Traceback (most recent call last): File "", line 1, in ? File "/home/was/sage/local/lib/python2.4/site-packages/sage /schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie ld.py", line 596, in gens raise RuntimeError, "Unable to compute the rank, hence ge nerators, with certainty (lower bound=%s). This could be bec ause Sha[2] is nontrivial."%C.rank() RuntimeError: Unable to compute the rank, hence generators, w ith certainty (lower bound=1). This could be because Sha[2] is nontrivial. sage: sage: E _16 = Elliptic Curve defined by y^2 = x^3 + 7*x + 4000000 ov er Rational Field sage: E.root_number() _17 = 1 sage: sage: E _18 = Elliptic Curve defined by y^2 = x^3 + 7*x + 4000000 ov er Rational Field sage: E.Ls E.Lseries E.Lseries_deriv_at1 E.Lseries_at1 E.Lseries_extended sage: E.Lseries_at1() *** sorry, anell for n >= %lu is not yet implemented. ------------------------------------------------------------ Traceback (most recent call last): File "", line 1, in ? File "/home/was/sage/local/lib/python2.4/site-packages/sage /schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie ld.py", line 978, in Lseries_at1 an = self.anlist(k) # list of C ints File "/home/was/sage/local/lib/python2.4/site-packages/sage /schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie ld.py", line 430, in anlist self.__anlist = [0] + [int(x) for x in E.ellan(n)] RuntimeError sage: sage: E _20 = Elliptic Curve defined by y^2 = x^3 + 7*x + 4000000 ov er Rational Field sage: E.tor E.torsion_order E.torsion_subgroup E.torsion_polynomial sage: E.torsion_subgroup() _21 = Abelian Group with invariants [] sage: E = EllipticCurve([0,-1]) sage: E _23 = Elliptic Curve defined by y^2 = x^3 -1 over Rational F ield sage: P = E([1,0]) sage: P _25 = (1, 0) sage: P + P _26 = 0 sage: E = EllipticCurve([0,0,1,-1,0]) sage: E _28 = Elliptic Curve defined by y^2 + y = x^3 - x over Ration al Field sage: P = E([0,0]) sage: P + P _30 = (1, 0) sage: P + P+ P _31 = (-1, -1) sage: P + P+ P + P _32 = (2, -3) sage: P + P+ P + P + P _33 = (1/4, -5/8) sage: P + P+ P + P + P + P _34 = (6, 14) sage: P + P+ P + P + P + P + P _35 = (-5/9, 8/27) sage: P + P+ P + P + P + P + P + P _36 = (21/25, -69/125) sage: was@form:~$ acpi Battery 1: charging, 61%, 01:36:23 until charged was@form:~$