CoCalc Public Fileswww / 168 / notes / 2005-10-12 / 2005-10-12-sage_log.txtOpen with one click!
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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[email protected]:~$ sage
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-------------------------------------------------------------
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-----------
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SAGE Version 0.7.11, Export Date: 2005-10-12-1029
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Distributed under the terms of the GNU General Public Licen
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se (GPL)
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IPython shell -- for help type <object>?, <object>??, %magi
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c, or help
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-------------------------------------------------------------
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-----------
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sage: E = EllipticCurve([0,-1])
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sage: E
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_2 = Elliptic Curve defined by y^2 = x^3 -1 over Rational F
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ield
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sage: E.gens(verbose=True)
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_3 = []
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sage: E.gens?
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sage: E.gens(verbose=True, only_use_mwrank=True)
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_5 = []
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sage: E = EllipticCurve([0,-1])
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sage: E.gens(verbose=True, only_use_mwrank=True)
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1 points of order 2:
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[1:0:1]
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Using 2-isogenous curve [0,-6,0,-3,0]
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-------------------------------------------------------
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First step, determining 1st descent Selmer groups
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-------------------------------------------------------
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After first local descent, rank bound = 0
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rk(S^{phi}(E'))= 1
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rk(S^{phi'}(E))= 1
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-------------------------------------------------------
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Second step, determining 2nd descent Selmer groups
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-------------------------------------------------------
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...skipping since we already know rank=0
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After second local descent, rank bound = 0
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rk(phi'(S^{2}(E)))= 1
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rk(phi(S^{2}(E')))= 1
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rk(S^{2}(E))= 1
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rk(S^{2}(E'))= 1
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Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
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-------------------------------------------------------
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1. E(Q)/phi(E'(Q))
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-------------------------------------------------------
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(c,d) =(3,3)
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(c',d')=(-6,-3)
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This component of the rank is 0
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-------------------------------------------------------
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2. E'(Q)/phi'(E(Q))
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-------------------------------------------------------
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This component of the rank is 0
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Summary of results:
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-------------------------------------------------------
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rank(E) = 0
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#E(Q)/2E(Q) = 2
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Information on III(E/Q):
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#III(E/Q)[phi'] = 1
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#III(E/Q)[2] = 1
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Information on III(E'/Q):
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#phi'(III(E/Q)[2]) = 1
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#III(E'/Q)[phi] = 1
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#III(E'/Q)[2] = 1
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Regulator (before saturation) = 1
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Searching for points (bound = 10)...done
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Regulator (after searching) = 1
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Saturating (bound = -1)...finished saturation (index was 0)
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Regulator (after saturation) = 1
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_7 = []
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sage: E = EllipticCurve([0,1,0,-1,0])
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sage: E
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_9 = Elliptic Curve defined by y^2 = x^3 + x^2 - x over Rat
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ional Field
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sage: E = EllipticCurve([0,0,1,-1,0])
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sage: E
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_11 = Elliptic Curve defined by y^2 + y = x^3 - x over Ration
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al Field
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sage: E.gens(verbose=True, only_use_mwrank=True)
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Basic pair: I=48, J=-432
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disc=255744
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2-adic index bound = 2
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By Lemma 5.1(a), 2-adic index = 1
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2-adic index = 1
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One (I,J) pair
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Looking for quartics with I = 48, J = -432
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Looking for Type 2 quartics:
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Trying positive a from 1 up to 1 (square a first...)
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(1,0,-6,4,1) --trivial
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Trying positive a from 1 up to 1 (...then non-square a)
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Finished looking for Type 2 quartics.
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Looking for Type 1 quartics:
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Trying positive a from 1 up to 2 (square a first...)
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(1,0,0,4,4) --nontrivial...(x:y:z) = (1 : 1 : 0)
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Point = [0:0:1]
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height = 0.051111408239968840235886099756942021609538
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202280854
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Rank of B=im(eps) increases to 1 (The previous point is on th
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e egg)
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Exiting search for Type 1 quartics after finding one which is
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globally soluble.
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Mordell rank contribution from B=im(eps) = 1
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Selmer rank contribution from B=im(eps) = 1
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Sha rank contribution from B=im(eps) = 0
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Mordell rank contribution from A=ker(eps) = 0
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Selmer rank contribution from A=ker(eps) = 0
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Sha rank contribution from A=ker(eps) = 0
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Regulator (before saturation) = 0.051111408239968840235886099
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756942021609538202280854
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Searching for points (bound = 10)...done
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Regulator (after searching) = 0.05111140823996884023588609975
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6942021609538202280854
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Saturating (bound = -1)...finished saturation (index was 1)
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Regulator (after saturation) = 0.0511114082399688402358860997
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56942021609538202280854
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_12 = [(0, 0)]
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sage: E = EllipticCurve([7,4000000])
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sage: E.gens(verbose=True, only_use_mwrank=True)
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Basic pair: I=-21, J=-108000000
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disc=-11664000000037044
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2-adic index bound = 4
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2-adic index = 4
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Two (I,J) pairs
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Looking for quartics with I = -21, J = -108000000
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Looking for Type 3 quartics:
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Trying positive a from 1 up to 137 (square a first...)
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Trying positive a from 1 up to 137 (...then non-square a)
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(20,-1,-309,806,-408) --nontrivial...locally soluble...no r
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ational point found (limit 8) --new (B) #1
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(70,99,165,318,80) --nontrivial...--equivalent to (B) #1
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Trying negative a from -1 down to -91
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Finished looking for Type 3 quartics.
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Looking for quartics with I = -336, J = -6912000000
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Looking for Type 3 quartics:
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Trying positive a from 1 up to 549 (square a first...)
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(1,0,0,16000,-28) --nontrivial...(x:y:z) = (1 : 1 : 0)
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Point = [0:2000:1]
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height = 5.529366426734982863340020597597267739712206
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3373968
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Rank of B=im(eps) increases to 1
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Trying positive a from 1 up to 549 (...then non-square a)
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(50,24,636,2056,-428) --nontrivial...not locally soluble (p
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= 2)
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(77,132,-1236,2384,-632) --nontrivial...not locally so
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luble (p = 2)
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(370,-736,1392,-560,-158) --nontrivial...not locally so
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luble (p = 2)
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Trying negative a from -1 down to -366
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(-107,172,4236,11504,9352) --nontrivial...not locally so
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luble (p = 2)
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(-158,-72,2124,4568,2900) --nontrivial...not locally so
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luble (p = 2)
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Finished looking for Type 3 quartics.
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Mordell rank contribution from B=im(eps) = 1
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Selmer rank contribution from B=im(eps) = 2
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Sha rank contribution from B=im(eps) = 1
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Mordell rank contribution from A=ker(eps) = 0
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Selmer rank contribution from A=ker(eps) = 0
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Sha rank contribution from A=ker(eps) = 0
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Warning: Selmer rank = 2 and program finds
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lower bound for rank = 1 which differs by an odd
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integer from the Selmer rank. Hence the rank must be 1 more
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than reported here. Try rerunning with a higher bound for
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quartic point search.
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Summary of results (all should be powers of 2):
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n0 = #E(Q)[2] = 1
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n1 = #E(Q)/2E(Q) >= 2
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n2 = #S^(2)(E/Q) = 4
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#III(E/Q)[2] <= 2
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1 <= rank <= selmer-rank = 2
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Regulator (before saturation) = 5.529366426734982863340020597
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5972677397122063373968
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Searching for points (bound = 10)...done
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Regulator (after searching) = 5.52936642673498286334002059759
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72677397122063373968
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Saturating (bound = -1)...-----------------------------------
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-------------------------
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Traceback (most recent call last):
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File "<console>", line 1, in ?
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File "/home/was/sage/local/lib/python2.4/site-packages/sage
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/schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie
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ld.py", line 596, in gens
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raise RuntimeError, "Unable to compute the rank, hence ge
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nerators, with certainty (lower bound=%s). This could be bec
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ause Sha[2] is nontrivial."%C.rank()
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RuntimeError: Unable to compute the rank, hence generators, w
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ith certainty (lower bound=1). This could be because Sha[2]
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is nontrivial.
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finished saturation (index was 1)
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Regulator (after saturation) = 5.5293664267349828633400205975
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972677397122063373968
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sage: E.gens()
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------------------------------------------------------------
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Traceback (most recent call last):
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File "<console>", line 1, in ?
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File "/home/was/sage/local/lib/python2.4/site-packages/sage
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/schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie
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ld.py", line 596, in gens
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raise RuntimeError, "Unable to compute the rank, hence ge
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nerators, with certainty (lower bound=%s). This could be bec
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ause Sha[2] is nontrivial."%C.rank()
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RuntimeError: Unable to compute the rank, hence generators, w
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ith certainty (lower bound=1). This could be because Sha[2]
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is nontrivial.
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sage:
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sage: E
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_16 = Elliptic Curve defined by y^2 = x^3 + 7*x + 4000000 ov
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er Rational Field
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sage: E.root_number()
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_17 = 1
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sage:
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sage: E
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_18 = Elliptic Curve defined by y^2 = x^3 + 7*x + 4000000 ov
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er Rational Field
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sage: E.Ls
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E.Lseries E.Lseries_deriv_at1
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E.Lseries_at1 E.Lseries_extended
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sage: E.Lseries_at1()
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*** sorry, anell for n >= %lu is not yet implemented.
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------------------------------------------------------------
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Traceback (most recent call last):
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File "<console>", line 1, in ?
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File "/home/was/sage/local/lib/python2.4/site-packages/sage
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/schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie
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ld.py", line 978, in Lseries_at1
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an = self.anlist(k) # list of C ints
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File "/home/was/sage/local/lib/python2.4/site-packages/sage
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/schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie
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ld.py", line 430, in anlist
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self.__anlist = [0] + [int(x) for x in E.ellan(n)]
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RuntimeError
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sage:
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sage: E
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_20 = Elliptic Curve defined by y^2 = x^3 + 7*x + 4000000 ov
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er Rational Field
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sage: E.tor
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E.torsion_order E.torsion_subgroup
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E.torsion_polynomial
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sage: E.torsion_subgroup()
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_21 = Abelian Group with invariants []
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sage: E = EllipticCurve([0,-1])
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sage: E
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_23 = Elliptic Curve defined by y^2 = x^3 -1 over Rational F
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ield
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sage: P = E([1,0])
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sage: P
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_25 = (1, 0)
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sage: P + P
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_26 = 0
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sage: E = EllipticCurve([0,0,1,-1,0])
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sage: E
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_28 = Elliptic Curve defined by y^2 + y = x^3 - x over Ration al Field
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sage: P = E([0,0])
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sage: P + P
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_30 = (1, 0)
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sage: P + P+ P
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_31 = (-1, -1)
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sage: P + P+ P + P
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_32 = (2, -3)
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sage: P + P+ P + P + P
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_33 = (1/4, -5/8)
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sage: P + P+ P + P + P + P
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_34 = (6, 14)
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sage: P + P+ P + P + P + P + P
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_35 = (-5/9, 8/27)
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sage: P + P+ P + P + P + P + P + P
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_36 = (21/25, -69/125)
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sage:
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Battery 1: charging, 61%, 01:36:23 until charged
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