CoCalc Public Fileswww / 168 / notes / 2005-10-12 / 2005-10-12-sage_log.txt
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1[email protected]:~$sage 2------------------------------------------------------------- 3----------- 4 SAGE Version 0.7.11, Export Date: 2005-10-12-1029 5 Distributed under the terms of the GNU General Public Licen 6se (GPL) 7 IPython shell -- for help type <object>?, <object>??, %magi 8c, or help 9------------------------------------------------------------- 10----------- 11 12sage: E = EllipticCurve([0,-1]) 13sage: E 14 _2 = Elliptic Curve defined by y^2 = x^3 -1 over Rational F 15ield 16sage: E.gens(verbose=True) 17 _3 = [] 18sage: E.gens? 19sage: E.gens(verbose=True, only_use_mwrank=True) 20 _5 = [] 21sage: E = EllipticCurve([0,-1]) 22sage: E.gens(verbose=True, only_use_mwrank=True) 23 241 points of order 2: 25[1:0:1] 26 27Using 2-isogenous curve [0,-6,0,-3,0] 28------------------------------------------------------- 29First step, determining 1st descent Selmer groups 30------------------------------------------------------- 31After first local descent, rank bound = 0 32rk(S^{phi}(E'))= 1 33rk(S^{phi'}(E))= 1 34 35------------------------------------------------------- 36Second step, determining 2nd descent Selmer groups 37------------------------------------------------------- 38...skipping since we already know rank=0 39After second local descent, rank bound = 0 40rk(phi'(S^{2}(E)))= 1 41rk(phi(S^{2}(E')))= 1 42rk(S^{2}(E))= 1 43rk(S^{2}(E'))= 1 44 45Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q)) 46------------------------------------------------------- 471. E(Q)/phi(E'(Q)) 48------------------------------------------------------- 49(c,d) =(3,3) 50(c',d')=(-6,-3) 51This component of the rank is 0 52------------------------------------------------------- 532. E'(Q)/phi'(E(Q)) 54------------------------------------------------------- 55This component of the rank is 0 56 57------------------------------------------------------- 58Summary of results: 59------------------------------------------------------- 60 rank(E) = 0 61 #E(Q)/2E(Q) = 2 62 63Information on III(E/Q): 64 #III(E/Q)[phi'] = 1 65 #III(E/Q)[2] = 1 66 67Information on III(E'/Q): 68 #phi'(III(E/Q)[2]) = 1 69 #III(E'/Q)[phi] = 1 70 #III(E'/Q)[2] = 1 71 72 73Regulator (before saturation) = 1 74Searching for points (bound = 10)...done 75Regulator (after searching) = 1 76Saturating (bound = -1)...finished saturation (index was 0) 77Regulator (after saturation) = 1 78 _7 = [] 79sage: E = EllipticCurve([0,1,0,-1,0]) 80sage: E 81 _9 = Elliptic Curve defined by y^2 = x^3 + x^2 - x over Rat 82ional Field 83sage: E = EllipticCurve([0,0,1,-1,0]) 84sage: E 85_11 = Elliptic Curve defined by y^2 + y = x^3 - x over Ration 86al Field 87sage: E.gens(verbose=True, only_use_mwrank=True) 88Basic pair: I=48, J=-432 89disc=255744 902-adic index bound = 2 91By Lemma 5.1(a), 2-adic index = 1 922-adic index = 1 93One (I,J) pair 94Looking for quartics with I = 48, J = -432 95Looking for Type 2 quartics: 96Trying positive a from 1 up to 1 (square a first...) 97(1,0,-6,4,1) --trivial 98Trying positive a from 1 up to 1 (...then non-square a) 99Finished looking for Type 2 quartics. 100Looking for Type 1 quartics: 101Trying positive a from 1 up to 2 (square a first...) 102(1,0,0,4,4) --nontrivial...(x:y:z) = (1 : 1 : 0) 103Point = [0:0:1] 104 height = 0.051111408239968840235886099756942021609538 105202280854 106Rank of B=im(eps) increases to 1 (The previous point is on th 107e egg) 108Exiting search for Type 1 quartics after finding one which is 109 globally soluble. 110Mordell rank contribution from B=im(eps) = 1 111Selmer rank contribution from B=im(eps) = 1 112Sha rank contribution from B=im(eps) = 0 113Mordell rank contribution from A=ker(eps) = 0 114Selmer rank contribution from A=ker(eps) = 0 115Sha rank contribution from A=ker(eps) = 0 116 117Regulator (before saturation) = 0.051111408239968840235886099 118756942021609538202280854 119Searching for points (bound = 10)...done 120Regulator (after searching) = 0.05111140823996884023588609975 1216942021609538202280854 122Saturating (bound = -1)...finished saturation (index was 1) 123Regulator (after saturation) = 0.0511114082399688402358860997 12456942021609538202280854 125_12 = [(0, 0)] 126sage: E = EllipticCurve([7,4000000]) 127sage: E.gens(verbose=True, only_use_mwrank=True) 128Basic pair: I=-21, J=-108000000 129disc=-11664000000037044 1302-adic index bound = 4 1312-adic index = 4 132Two (I,J) pairs 133Looking for quartics with I = -21, J = -108000000 134Looking for Type 3 quartics: 135Trying positive a from 1 up to 137 (square a first...) 136Trying positive a from 1 up to 137 (...then non-square a) 137(20,-1,-309,806,-408) --nontrivial...locally soluble...no r 138ational point found (limit 8) --new (B) #1 139(70,99,165,318,80) --nontrivial...--equivalent to (B) #1 140Trying negative a from -1 down to -91 141Finished looking for Type 3 quartics. 142Looking for quartics with I = -336, J = -6912000000 143Looking for Type 3 quartics: 144Trying positive a from 1 up to 549 (square a first...) 145(1,0,0,16000,-28) --nontrivial...(x:y:z) = (1 : 1 : 0) 146Point = [0:2000:1] 147 height = 5.529366426734982863340020597597267739712206 1483373968 149Rank of B=im(eps) increases to 1 150Trying positive a from 1 up to 549 (...then non-square a) 151(50,24,636,2056,-428) --nontrivial...not locally soluble (p 152 = 2) 153(77,132,-1236,2384,-632) --nontrivial...not locally so 154luble (p = 2) 155(370,-736,1392,-560,-158) --nontrivial...not locally so 156luble (p = 2) 157Trying negative a from -1 down to -366 158(-107,172,4236,11504,9352) --nontrivial...not locally so 159luble (p = 2) 160(-158,-72,2124,4568,2900) --nontrivial...not locally so 161luble (p = 2) 162Finished looking for Type 3 quartics. 163Mordell rank contribution from B=im(eps) = 1 164Selmer rank contribution from B=im(eps) = 2 165Sha rank contribution from B=im(eps) = 1 166Mordell rank contribution from A=ker(eps) = 0 167Selmer rank contribution from A=ker(eps) = 0 168Sha rank contribution from A=ker(eps) = 0 169 170Warning: Selmer rank = 2 and program finds 171lower bound for rank = 1 which differs by an odd 172integer from the Selmer rank. Hence the rank must be 1 more 173than reported here. Try rerunning with a higher bound for 174quartic point search. 175 176Summary of results (all should be powers of 2): 177 178n0 = #E(Q)[2] = 1 179n1 = #E(Q)/2E(Q) >= 2 180n2 = #S^(2)(E/Q) = 4 181#III(E/Q)[2] <= 2 182 1831 <= rank <= selmer-rank = 2 184 185 186Regulator (before saturation) = 5.529366426734982863340020597 1875972677397122063373968 188Searching for points (bound = 10)...done 189Regulator (after searching) = 5.52936642673498286334002059759 19072677397122063373968 191Saturating (bound = -1)...----------------------------------- 192------------------------- 193Traceback (most recent call last): 194 File "<console>", line 1, in ? 195 File "/home/was/sage/local/lib/python2.4/site-packages/sage 196/schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie 197ld.py", line 596, in gens 198 raise RuntimeError, "Unable to compute the rank, hence ge 199nerators, with certainty (lower bound=%s). This could be bec 200ause Sha[2] is nontrivial."%C.rank() 201RuntimeError: Unable to compute the rank, hence generators, w 202ith certainty (lower bound=1). This could be because Sha[2] 203is nontrivial. 204 205finished saturation (index was 1) 206Regulator (after saturation) = 5.5293664267349828633400205975 207972677397122063373968 208sage: E.gens() 209------------------------------------------------------------ 210Traceback (most recent call last): 211 File "<console>", line 1, in ? 212 File "/home/was/sage/local/lib/python2.4/site-packages/sage 213/schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie 214ld.py", line 596, in gens 215 raise RuntimeError, "Unable to compute the rank, hence ge 216nerators, with certainty (lower bound=%s). This could be bec 217ause Sha[2] is nontrivial."%C.rank() 218RuntimeError: Unable to compute the rank, hence generators, w 219ith certainty (lower bound=1). This could be because Sha[2] 220is nontrivial. 221 222sage: 223sage: E 224_16 = Elliptic Curve defined by y^2 = x^3 + 7*x + 4000000 ov 225er Rational Field 226sage: E.root_number() 227_17 = 1 228sage: 229sage: E 230_18 = Elliptic Curve defined by y^2 = x^3 + 7*x + 4000000 ov 231er Rational Field 232sage: E.Ls 233E.Lseries E.Lseries_deriv_at1 234E.Lseries_at1 E.Lseries_extended 235sage: E.Lseries_at1() 236 237 *** sorry, anell for n >= %lu is not yet implemented. 238------------------------------------------------------------ 239Traceback (most recent call last): 240 File "<console>", line 1, in ? 241 File "/home/was/sage/local/lib/python2.4/site-packages/sage 242/schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie 243ld.py", line 978, in Lseries_at1 244 an = self.anlist(k) # list of C ints 245 File "/home/was/sage/local/lib/python2.4/site-packages/sage 246/schemes/hypersurfaces/plane_curves/elliptic/ell_rational_fie 247ld.py", line 430, in anlist 248 self.__anlist = [0] + [int(x) for x in E.ellan(n)] 249RuntimeError 250 251sage: 252sage: E 253_20 = Elliptic Curve defined by y^2 = x^3 + 7*x + 4000000 ov 254er Rational Field 255sage: E.tor 256E.torsion_order E.torsion_subgroup 257E.torsion_polynomial 258sage: E.torsion_subgroup() 259_21 = Abelian Group with invariants [] 260sage: E = EllipticCurve([0,-1]) 261sage: E 262_23 = Elliptic Curve defined by y^2 = x^3 -1 over Rational F 263ield 264sage: P = E([1,0]) 265sage: P 266_25 = (1, 0) 267sage: P + P 268_26 = 0 269sage: E = EllipticCurve([0,0,1,-1,0]) 270sage: E 271_28 = Elliptic Curve defined by y^2 + y = x^3 - x over Ration al Field 272sage: P = E([0,0]) 273sage: P + P 274_30 = (1, 0) 275sage: P + P+ P 276_31 = (-1, -1) 277sage: P + P+ P + P 278_32 = (2, -3) 279sage: P + P+ P + P + P 280_33 = (1/4, -5/8) 281sage: P + P+ P + P + P + P 282_34 = (6, 14) 283sage: P + P+ P + P + P + P + P 284_35 = (-5/9, 8/27) 285sage: P + P+ P + P + P + P + P + P 286_36 = (21/25, -69/125) 287sage: 288[email protected]:~$ acpi
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