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Loading startup file "/home/was/magma/local/init.m" Loading "bsd.m" 258E Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 44124*x + 3549153 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 + 15*x^3 - 7148007*x^2 + 10349330877*x - 4219023681882 3-division polynomial factors into x^4 + 15*x^3 - 7148007*x^2 + 10349330877*x - 4219023681882 A is defined by x^8 + 63*x^7 + 20773718028*x^6 + 205292179489401*x^5 + 849269495069545176*x^4 + 1853986799451076523037*x^3 + 2216653022569197673770300*x^2 + 1349086467784303120414036923*x - 1316798807317184063767090248081 Using better representation of A, given by x^8 + 4*x^7 + 7*x^6 + 7*x^5 + 28*x^4 + 49*x^3 + 136*x^2 + 112*x - 20 Bad prime 2 --> c_2 = 2 Bad prime 3 --> c_3 = 1 Bad prime 43 --> c_43 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 + 4*x^7 + 7*x^6 + 7*x^5 + 28*x^4 + 49*x^3 + 136*x^2 + 112*x - 20 over the Rational Field S = [ <3, 2, 1>, <3, 3, 2> ] Computing class group... Class group has invariants [ 2 ] Kernel of norm has invariants [ 2 ] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 2, 6 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 1 2) Images of basis elements of V: (0 1 1 0 1 1 0 0 1 1) (0 1 1 0 0 0 2 0 1 2) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 378G Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 + 3967*x + 38449 over Rational Field 3-division polynomial = x^4 - x^3 + 7935*x^2 + 153797*x - 5285468 3-division polynomial factors into x^4 - x^3 + 7935*x^2 + 153797*x - 5285468 A is defined by x^8 + 5*x^7 + 319507*x^6 + 85103048*x^5 + 193220165008*x^4 + 498197300000*x^3 + 386853855144832*x^2 - 3322195876880896*x - 3153156303457816942592 Using better representation of A, given by x^8 + 4*x^7 + 7*x^6 + 7*x^5 + 4*x^4 + x^3 - 16*x^2 - 16*x - 4 Bad prime 2 --> c_2 = 5 Bad prime 3 --> c_3 = 1 Bad prime 7 --> c_7 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 + 4*x^7 + 7*x^6 + 7*x^5 + 4*x^4 + x^3 - 16*x^2 - 16*x - 4 over the Rational Field S = [ <3, 2, 1>, <3, 3, 2> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 2, 6 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 2 0 0 1 0 1 1 2 1) (0 0 0 0 2 0 2 0 1 1) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 594F Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 83*x + 325 over Rational Field 3-division polynomial = x^4 - x^3 - 165*x^2 + 1301*x - 2594 3-division polynomial factors into x^4 - x^3 - 165*x^2 + 1301*x - 2594 A is defined by x^8 + 5*x^7 + 2365*x^6 + 45002*x^5 + 542476*x^4 + 2132294*x^3 - 2112056*x^2 - 27970960*x - 2236615328 Using better representation of A, given by x^8 + 4*x^7 + 7*x^6 + 7*x^5 - 14*x^4 - 35*x^3 + 18*x^2 + 36*x - 18 Bad prime 2 --> c_2 = 1 Bad prime 3 --> c_3 = 1 Bad prime 11 --> c_11 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 + 4*x^7 + 7*x^6 + 7*x^5 - 14*x^4 - 35*x^3 + 18*x^2 + 36*x - 18 over the Rational Field S = [ <3, 2, 1>, <3, 3, 1>, <3, 3, 1> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 1 The group of principal ideals mod cubes supported in S has rank 1 ker N on K(S, 3) has dimension 3 ker(N) on A(S, 3) has dimension 3 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 3, 3, 2 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 1 0 0 0 2) Images of basis elements of V: (0 1 1 2 0 2 1 2 0 2 1 2) (0 2 2 1 0 2 0 0 0 2 2 1) (0 1 1 0 1 0 0 2 2 2 0 2) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 600G Compute 3-Selmer group of Elliptic Curve defined by y^2 = x^3 - x^2 - 5833*x + 207037 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 - 12*x^3 - 944946*x^2 + 603719892*x - 76221404919 3-division polynomial factors into x^4 - 12*x^3 - 944946*x^2 + 603719892*x - 76221404919 A is defined by x^8 + 1196100000*x^6 + 121068902812500000*x^4 - 1221473269018547534179687500000000 Using better representation of A, given by x^8 - 4*x^7 + 4*x^6 - 6*x^5 - 6*x^3 + 4*x^2 - 4*x + 1 Bad prime 2 --> c_2 = 2 Bad prime 3 --> c_3 = 1 Bad prime 5 --> c_5 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 - 4*x^7 + 4*x^6 - 6*x^5 - 6*x^3 + 4*x^2 - 4*x + 1 over the Rational Field S = [ <3, 2, 1>, <3, 3, 2> ] Computing class group... Class group has invariants [ 2 ] Kernel of norm has invariants [ 2 ] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 2, 6 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 2 0 0 1 0 0 0 0 0) (0 1 0 0 0 0 0 2 1 2) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 612D Compute 3-Selmer group of Elliptic Curve defined by y^2 = x^3 - 14592*x + 679412 over Rational Field 3-division polynomial = x^4 - 29184*x^2 + 2717648*x - 70975488 3-division polynomial factors into x^4 - 29184*x^2 + 2717648*x - 70975488 A is defined by x^8 + 5435296*x^6 + 23417416224*x^4 - 45697948550671534848 Using better representation of A, given by x^8 + 3*x^7 - 8*x^6 - 32*x^5 - 15*x^4 + 61*x^3 + 154*x^2 + 156*x + 88 Bad prime 2 --> c_2 = 1 Bad prime 3 --> c_3 = 2 Bad prime 17 --> c_17 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 + 3*x^7 - 8*x^6 - 32*x^5 - 15*x^4 + 61*x^3 + 154*x^2 + 156*x + 88 over the Rational Field S = [ <3, 6, 1>, <3, 1, 2> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 6, 2 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 1 2 0 0 0) Images of basis elements of V: (0 0 0 0 2 0 1 0 0 0) (0 0 0 0 1 2 0 0 0 2) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 626B Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y + y = x^3 - 2210*x + 39796 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 + 3*x^3 - 357939*x^2 + 116045865*x - 10589659578 3-division polynomial factors into x^4 + 3*x^3 - 357939*x^2 + 116045865*x - 10589659578 A is defined by x^8 + 99*x^7 + 231559101*x^6 + 528897092535*x^5 + 505829444517891*x^4 + 254228453552025945*x^3 + 69313808624597725455*x^2 + 9444550203976160485197*x - 624297299730378826990608 Using better representation of A, given by x^8 - 72*x^4 - 340*x^2 - 432 Bad prime 2 --> c_2 = 1 Bad prime 313 --> c_313 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 - 72*x^4 - 340*x^2 - 432 over the Rational Field S = [ <3, 2, 1>, <3, 3, 1>, <3, 3, 1> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 1 The group of principal ideals mod cubes supported in S has rank 1 ker N on K(S, 3) has dimension 3 ker(N) on A(S, 3) has dimension 3 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 2, 3, 3 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 1 0 0 0 2 0 0 0 1) Images of basis elements of V: (0 0 0 0 0 2 2 1 0 1 1 0) (0 1 0 1 0 1 0 0 0 2 0 2) (0 2 0 1 1 0 1 1 2 0 2 0) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 658A Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 117008*x + 18214144 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 + 15*x^3 - 18955296*x^2 + 53112443904*x - 29742765539328 3-division polynomial factors into x^4 + 15*x^3 - 18955296*x^2 + 53112443904*x - 29742765539328 A is defined by x^8 - 45*x^7 + 106423919766*x^6 + 1433274504760512*x^5 + 931614206197424564736*x^4 + 26911958544912375447552*x^3 - 39170148882116291289261932544*x^2 - 164954412918416642925289285877760*x - 70578959722056691349001990581376836960256 Using better representation of A, given by x^8 - 12*x^4 - 29*x^2 - 12 Bad prime 2 --> c_2 = 2 Bad prime 7 --> c_7 = 1 Bad prime 47 --> c_47 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 - 12*x^4 - 29*x^2 - 12 over the Rational Field S = [ <3, 2, 1>, <3, 3, 2> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 2, 6 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 2 2 0 0 0 0 1 2 1) (0 1 1 0 0 2 1 0 1 1) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 681B Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 1154*x - 15345 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 + 15*x^3 - 186948*x^2 - 44746020*x - 3080260467 3-division polynomial factors into x^4 + 15*x^3 - 186948*x^2 - 44746020*x - 3080260467 A is defined by x^8 - 45*x^7 - 87527736*x^6 + 143171086761*x^5 - 162413660744694*x^4 + 35188552298403225*x^3 + 21427656196709810664*x^2 - 9373868685034969903938*x - 377289471384781392440720331 Using better representation of A, given by x^8 - 6*x^4 + 235*x^2 - 3 Bad prime 3 --> c_3 = 2 Bad prime 227 --> c_227 = 2 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 - 6*x^4 + 235*x^2 - 3 over the Rational Field S = [ <3, 2, 1>, <3, 3, 2> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 2, 6 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 0 0 0 0 0 0 0 0 2) (0 0 0 0 0 0 0 0 0 0) Basis of inverse image in V: (1 0) (0 1) Local restriction at 3 -> dimension = 2 Restriction to kernel of map to B^*/(B^*)^3 -> dimension = 2 735B Compute 3-Selmer group of Elliptic Curve defined by y^2 + y = x^3 - x^2 - 15206*x - 1184338 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 - 12*x^3 - 2463372*x^2 - 3453528879*x - 495322880895 3-division polynomial factors into x^4 - 12*x^3 - 2463372*x^2 - 3453528879*x - 495322880895 A is defined by x^8 + 108*x^7 - 6936613551*x^6 - 561865973193*x^5 + 8549723154050514135*x^4 + 461685732985952154351*x^3 + 9349139548968524101818*x^2 + 84142268382327840309159*x - 6091507522308628991168899084257215103 Using better representation of A, given by x^8 - 4*x^7 + 4*x^6 - 42*x^5 + 105*x^4 + 147*x^3 - 126*x^2 - 441*x - 315 Bad prime 3 --> c_3 = 1 Bad prime 5 --> c_5 = 2 Bad prime 7 --> c_7 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 - 4*x^7 + 4*x^6 - 42*x^5 + 105*x^4 + 147*x^3 - 126*x^2 - 441*x - 315 over the Rational Field S = [ <3, 2, 1>, <3, 3, 2> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 6, 2 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 2 1 0 0 0 2 0 1 0) (0 0 0 0 2 2 1 0 1 1) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 738B Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 1575*x + 751869 over Rational Field 3-division polynomial = x^4 - x^3 - 3150*x^2 + 3007476*x - 1578744 3-division polynomial factors into x^4 - x^3 - 3150*x^2 + 3007476*x - 1578744 A is defined by x^8 + x^7 + 6010227*x^6 + 16989345*x^5 + 10166294603061*x^4 + 10730886782859*x^3 - 7992813506027031*x^2 - 3817850585699116845*x - 8592735171052284654416274 Using better representation of A, given by x^8 - 4*x^7 + 7*x^6 - 7*x^5 + x^4 + 5*x^3 - 32*x^2 + 29*x - 8 Bad prime 2 --> c_2 = 1 Bad prime 3 --> c_3 = 2 Bad prime 41 --> c_41 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 - 4*x^7 + 7*x^6 - 7*x^5 + x^4 + 5*x^3 - 32*x^2 + 29*x - 8 over the Rational Field S = [ <3, 1, 1>, <3, 1, 1>, <3, 6, 1> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 1 The group of principal ideals mod cubes supported in S has rank 1 ker N on K(S, 3) has dimension 3 ker(N) on A(S, 3) has dimension 3 Finding local image for p = 3 ... dim E(Q_3)[3] = 1 Defining polynomial of A factors into degrees [ 1, 1, 6 ] Local image of 3-torsion is (0 0 0 0 0 0 0 0 0 1 2 2) Total local image has dimension 2 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 0 1 2 0) (0 0 0 0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 0 0 0 0 1 0 1 2 2 1 2) (0 2 0 1 0 2 0 2 0 2 0 1) (1 1 2 2 0 1 0 1 0 2 2 1) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 742F Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 81*x + 11797 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 - 9*x^3 - 13041*x^2 + 34400781*x - 91574064 3-division polynomial factors into x^4 - 9*x^3 - 13041*x^2 + 34400781*x - 91574064 A is defined by x^8 + 135*x^7 + 68633163*x^6 + 7400204388*x^5 + 1331432596220472*x^4 + 109765965547497276*x^3 - 35679444518442476160*x^2 - 155025747907272225275904*x - 146536985386406001813885222912 Using better representation of A, given by x^8 - 4*x^7 + 7*x^6 - 7*x^5 - 38*x^4 + 83*x^3 + 82*x^2 - 124*x - 116 Bad prime 2 --> c_2 = 2 Bad prime 7 --> c_7 = 2 Bad prime 53 --> c_53 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 - 4*x^7 + 7*x^6 - 7*x^5 - 38*x^4 + 83*x^3 + 82*x^2 - 124*x - 116 over the Rational Field S = [ <3, 8, 1> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 8 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 2 2 0 1 0 2 0 1) (0 0 0 0 0 2 0 0 1) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 777B Compute 3-Selmer group of Elliptic Curve defined by y^2 + y = x^3 - x^2 - 2531950*x + 1551713040 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 - 12*x^3 - 410175900*x^2 + 4524795225369*x - 14033930130743607 3-division polynomial factors into x^4 - 12*x^3 - 410175900*x^2 + 4524795225369*x - 14033930130743607 A is defined by x^8 + 108*x^7 + 9044668344609*x^6 + 732618135637767*x^5 + 4687431101214003323829*x^4 + 253120389334521446561427*x^3 + 5125683377735696126930832*x^2 + 46131134176983158093678895*x - 1830981543974158056615465972509967563081277 Using better representation of A, given by x^8 - 6*x^4 - 29*x^2 - 3 Bad prime 3 --> c_3 = 2 Bad prime 7 --> c_7 = 1 Bad prime 37 --> c_37 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 - 6*x^4 - 29*x^2 - 3 over the Rational Field S = [ <3, 2, 1>, <3, 3, 2> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 2, 6 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 2 2 0 0 0 0 2 1 0) (0 1 1 0 0 0 0 2 1 0) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 780B Compute 3-Selmer group of Elliptic Curve defined by y^2 = x^3 - x^2 + 195*x - 195975 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 - 12*x^3 + 31590*x^2 - 571463100*x + 1631228625 3-division polynomial factors into x^4 - 12*x^3 + 31590*x^2 - 571463100*x + 1631228625 A is defined by x^8 - 1142547552*x^6 + 367158397425300000*x^4 - 11233774066659545139090007500000000 Using better representation of A, given by x^8 - 96*x^4 + 112*x^2 - 768 Bad prime 2 --> c_2 = 1 Bad prime 3 --> c_3 = 1 Bad prime 5 --> c_5 = 5 Bad prime 13 --> c_13 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 - 96*x^4 + 112*x^2 - 768 over the Rational Field S = [ <3, 2, 1>, <3, 3, 2> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 2, 6 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 0 0 0 0 0 0 1 2 0) (0 1 1 0 0 0 0 0 0 1) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 819D Compute 3-Selmer group of Elliptic Curve defined by y^2 + y = x^3 + 22857*x + 4273542 over Rational Field 3-division polynomial = x^4 + 45714*x^2 + 17094169*x - 174147483 3-division polynomial factors into x^4 + 45714*x^2 + 17094169*x - 174147483 A is defined by x^8 + 4*x^7 + 34188345*x^6 + 102565021*x^5 + 360580980883351*x^4 + 721161790825005*x^3 + 540871311067186*x^2 + 180290432748853*x - 10834879276444097462551601949 Using better representation of A, given by x^8 + x^7 - 3*x^6 - 2*x^5 + 4*x^4 - 3*x^3 - 5*x^2 + 7*x - 3 Bad prime 3 --> c_3 = 2 Bad prime 7 --> c_7 = 1 Bad prime 13 --> c_13 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 + x^7 - 3*x^6 - 2*x^5 + 4*x^4 - 3*x^3 - 5*x^2 + 7*x - 3 over the Rational Field S = [ <3, 1, 1>, <3, 1, 1>, <3, 6, 1> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 1 The group of principal ideals mod cubes supported in S has rank 1 ker N on K(S, 3) has dimension 3 ker(N) on A(S, 3) has dimension 3 Finding local image for p = 3 ... dim E(Q_3)[3] = 1 Defining polynomial of A factors into degrees [ 1, 1, 6 ] Local image of 3-torsion is (0 0 0 0 0 0 0 0 0 1 1 1) Total local image has dimension 2 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 0 1 1 0) (0 0 0 0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 0 0 0 0 1 2 1 2 2 1 1) (0 1 0 2 0 1 2 1 0 1 0 0) (1 0 2 0 0 1 2 1 1 2 1 2) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 850I Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 + 195*x + 2197 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 - 9*x^3 + 31671*x^2 + 6407181*x - 98003844 3-division polynomial factors into x^4 - 9*x^3 + 31671*x^2 + 6407181*x - 98003844 A is defined by x^8 + 135*x^7 + 13249575*x^6 + 5301288000*x^5 + 56881901700000*x^4 + 4546451182950000*x^3 + 4348665848862000000*x^2 - 1149337119486960000000*x - 288346213510789046400000000 Using better representation of A, given by x^8 + 4*x^7 + 7*x^6 + 7*x^5 - 50*x^4 - 107*x^3 - 218*x^2 - 164*x - 284 Bad prime 2 --> c_2 = 4 Bad prime 5 --> c_5 = 1 Bad prime 17 --> c_17 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 + 4*x^7 + 7*x^6 + 7*x^5 - 50*x^4 - 107*x^3 - 218*x^2 - 164*x - 284 over the Rational Field S = [ <3, 8, 1> ] Computing class group... Class group has invariants [ 2 ] Kernel of norm has invariants [ 2 ] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 8 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 1 1 0 1 1 1 1 2) (0 1 1 0 1 0 1 1 2) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 978C Compute 3-Selmer group of Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 2188119*x - 1243572651 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 + 15*x^3 - 354475278*x^2 - 3626257850316*x - 10484658693036792 3-division polynomial factors into x^4 + 15*x^3 - 354475278*x^2 - 3626257850316*x - 10484658693036792 A is defined by x^8 - 45*x^7 - 7248793708863*x^6 + 502880236206420159*x^5 - 89621759803519313216001*x^4 + 223573533813326056801608441*x^3 + 58019180520216464324809720855371*x^2 - 910836270603941235568482424276644939*x - 470720444477531147694347301119254535308554444 Using better representation of A, given by x^8 + 4*x^7 + 7*x^6 + 7*x^5 - 5*x^4 - 17*x^3 - 70*x^2 - 61*x - 22 Bad prime 2 --> c_2 = 1 Bad prime 3 --> c_3 = 2 Bad prime 163 --> c_163 = 1 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 + 4*x^7 + 7*x^6 + 7*x^5 - 5*x^4 - 17*x^3 - 70*x^2 - 61*x - 22 over the Rational Field S = [ <3, 2, 1>, <3, 3, 2> ] Computing class group... Class group has invariants [] Kernel of norm has invariants [] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 2, 6 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 1 1) Images of basis elements of V: (0 2 0 0 0 0 2 2 1 2) (0 2 0 0 1 1 0 2 1 2) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 980I Compute 3-Selmer group of Elliptic Curve defined by y^2 = x^3 + 1568*x - 72716 over Rational Field 3-division polynomial is not integral; scaling equation of E. 3-division polynomial = x^4 + 254016*x^2 - 212039856*x - 5377010688 3-division polynomial factors into x^4 + 254016*x^2 - 212039856*x - 5377010688 A is defined by x^8 - 424079712*x^6 + 56044400086755360*x^4 - 261747898423692016957874374060800 Using better representation of A, given by x^8 + 4*x^7 - 5*x^6 - 41*x^5 + 46*x^4 + 121*x^3 - 230*x^2 - 140*x + 280 Bad prime 2 --> c_2 = 1 Bad prime 5 --> c_5 = 1 Bad prime 7 --> c_7 = 2 Bad primes for 3-descent: [ 3 ] KS3 with K = Number Field with defining polynomial x^8 + 4*x^7 - 5*x^6 - 41*x^5 + 46*x^4 + 121*x^3 - 230*x^2 - 140*x + 280 over the Rational Field S = [ <3, 8, 1> ] Computing class group... Class group has invariants [ 2 ] Kernel of norm has invariants [ 2 ] 3-torsion subgroup has rank 0 Computing unit group... Unit group has invariants [ 2, 0, 0, 0, 0 ] Setting up norm map on units... Computing kernel of norm map... Kernel of norm has invariants [ 2, 0, 0 ] Ker N on U/U^3 has rank 2 Ker N on K(empty, 3) has dimension 2 Dimension of kernel of N on F^S = 0 The group of principal ideals mod cubes supported in S has rank 0 ker N on K(S, 3) has dimension 2 ker(N) on A(S, 3) has dimension 2 Finding local image for p = 3 ... dim E(Q_3)[3] = 0 Defining polynomial of A factors into degrees [ 8 ] Local image of 3-torsion is trivial Total local image has dimension 1 Need additional generator Basis of local image: (0 0 0 0 0 0 0 0 1) Images of basis elements of V: (0 0 0 0 0 1 1 1 1) (0 1 2 1 2 2 2 2 0) Basis of inverse image in V: Local restriction at 3 -> dimension = 0 Total time: 343.389 seconds, Total memory usage: 4.88MB