[was@modular was]$ [was@modular was]$ Magma V2.8-1 Mon Sep 3 2001 11:52:46 on modular [Seed = 1] Linked at: Fri Aug 10 2001 15:27:36 Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > E := EC([0,1267645681,0,-165381887847758400,0]); EC( s: [ 0, 1267645681, 0, -165381887847758400, 0 ] ) In file "/home/was/magma/local/init.m", line 123, column 24: >> return EllipticCurve(CremonaDatabase(),s); ^ Runtime error in 'EllipticCurve': Bad argument types Argument types given: DB, SeqEnum[RngIntElt] > E := EllipticCurve([0,1267645681,0,-165381887847758400,0]); > E; Elliptic Curve defined by y^2 = x^3 + 1267645681*x^2 - 165381887847758400*x over Rational Field > factor(Conductor(E)); [ <2, 1>, <3, 1>, <5, 1>, <7, 1>, <13, 1>, <167, 1>, <197, 1>, <223, 1> ] 1 > Conductor(E); 20028582210 > TraceOfFrobenius(BaseExtend(E,GF(2))); > TraceOfFrobenius(BaseExtend(E,GF(2))); >> TraceOfFrobenius(BaseExtend(E,GF(2))); ^ Runtime error in 'BaseExtend': Coercion of equations is not possible > E; Elliptic Curve defined by y^2 + x*y = x^3 - 43813984085442270*x + 3449557430431948262718900 over Rational Field > MinimalModel(E); Elliptic Curve defined by y^2 + x*y = x^3 - 43813984085442270*x + 3449557430431948262718900 over Rational Field > E := $1; > TraceOfFrobenius(ChangeBase(E,GF(2))); >> TraceOfFrobenius(ChangeBase(E,GF(2))); ^ Runtime error in 'ChangeBase': Bad argument types Argument types given: CrvEll, FldFin > TraceOfFrobenius(ChangeBase(E,GF(2))); > TraceOfFrobenius(ChangeBase(E,GF(2))); >> TraceOfFrobenius(ChangeBase(E,GF(2))); ^ Runtime error in 'ChangeBase': Bad argument types Argument types given: CrvEll, FldFin > TraceOfFrobenius(ChangeRing(E,GF(2))); >> Trace(ChangeRing(E,GF(2))); ^ Runtime error in 'TraceOfFrobenius': Bad argument types Argument types given: Crv > > Trace(ChangeRing(E,GF(2))); >> Trace(ChangeRing(E,GF(2))); ^ Runtime error in 'Trace': Bad argument types Argument types given: Crv > qEigenform(E,10); q + q^2 + q^3 + q^4 + q^5 + q^6 + q^7 + q^8 + q^9 + O(q^10) > E; Elliptic Curve defined by y^2 + x*y = x^3 - 43813984085442270*x + 3449557430431948262718900 over Rational Field > Type(E); CrvEll > Type(ChangeRing(E,GF(2))); Crv > ChangeRing(E,GF(2)); Curve over GF(2) defined by $.1^3 + $.1*$.2*$.3 + $.2^2*$.3 > ReductionType(E,2); Split multiplicative > ReductionType(E,3); Split multiplicative > ReductionType(E,5); Split multiplicative > ReductionType(E,7); Split multiplicative > ReductionType(E,13); Split multiplicative > ReductionType(E,167); Nonsplit multiplicative > ReductionType(E,197); Split multiplicative > ReductionType(E,223); Nonsplit multiplicative > E; Elliptic Curve defined by y^2 + x*y = x^3 - 43813984085442270*x + 3449557430431948262718900 over Rational Field > RankBounds; Intrinsic 'RankBounds' Signatures: ( E) -> RngIntElt, RngIntElt [ Bound: RngIntElt ] ( E) -> RngIntElt, RngIntElt [ Bound: RngIntElt ] The lower and upper bounds of the rank of the Mordell-Weil group of E; E must be defined over Q > time RankBounds(e : Bound := 1); >> time RankBounds(e : Bound := 1); ^ User error: Identifier 'e' has not been declared or assigned > time RankBounds(E : Bound := 1); >> time RankBounds(E : Bound := 1); ^ Runtime error in 'RankBounds': Bound must be at least 2 > time RankBounds(E : Bound := 2); Warning: rank computed (0) is only a lower bound (It may still be correct, though) 0 5 Time: 0.330 > time RankBounds(E : Bound := 3); 0 5 Time: 0.000 > time RankBounds(E : Bound := 5); 0 5 Time: 0.000 > time RankBounds(E : Bound := 10); 0 5 Time: 0.000 > time Rank(E : Bound := 3); Warning: rank computed (0) is only a lower bound (It may still be correct, though) 0 Time: 0.000 > time Rank(E : Bound := 2); Warning: rank computed (0) is only a lower bound (It may still be correct, though) 0 Time: 0.000 > time Rank(E : Bound := 4); Warning: rank computed (0) is only a lower bound (It may still be correct, though) 0 Time: 0.000 > MordellWeilGroup; Intrinsic 'MordellWeilGroup' Signatures: ( E) -> GrpAb, Map [ Bound: RngIntElt, HeightBound: RngElt ] ( E) -> GrpAb, Map [ Bound: RngIntElt, HeightBound: RngElt ] The Mordell-Weil group of E; E must be defined over Q > time MordellWeilGroup(E : Bound := 4, HeightBound := 1); Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 1.080 > time MordellWeilGroup(E : Bound := 4, HeightBound := 2); Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 0.000 > time MordellWeilGroup(E : HeightBound := 2); Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 0.000 > time MordellWeilGroup(E : HeightBound := 4); Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 0.010 > time MordellWeilGroup(E : HeightBound := 3); Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 0.000 > E := EllipticCurve([0,1267645681,0,-165381887847758400,0]); > time MordellWeilGroup(E : HeightBound := 3); Warning: rank computed (0) is only a lower bound (It may still be correct, though) Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 5.450 > E := EllipticCurve([0,1267645681,0,-165381887847758400,0]); > time MordellWeilGroup(E : HeightBound := 5); Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 0.000 > E := EllipticCurve([0,1267645681,0,-165381887847758400,0]); > time MordellWeilGroup(E : HeightBound := 5); Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 0.000 > E := EllipticCurve([0,1267645681,0,-165381887847758400,0]); > time MordellWeilGroup(E : HeightBound := 7); Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 0.000 > quit; Total time: 9.650 seconds [was@modular was]$ me Magma V2.8-1 Mon Sep 3 2001 12:05:36 on modular [Seed = 1] Linked at: Fri Aug 10 2001 15:27:36 Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > E := EllipticCurve([0,1267645681,0,-165381887847758400,0]); > time MordellWeilGroup(E : HeightBound := 7); Warning: rank computed (0) is only a lower bound (It may still be correct, though) Abelian Group isomorphic to Z/2 + Z/4 Defined on 2 generators Relations: 2*$.1 = 0 4*$.2 = 0 Time: 5.460 > Sqrt(Conductor(E)); 141522.3735315374082847946013 > quit; Total time: 7.570 seconds [was@modular was]$ exit exit Process magma finished