> disc:=[[a,b]: a in [-50..50], b in [-50..50] | 4*a^3-27*b^2 ne 0]; > e := [: d in disc]; >> e := [: d in disc]; ^ Runtime error in 'EllipticCurve': Curve is not non-singular > disc:=[[a,b]: a in [-50..50], b in [-50..50] | 4*a^3+27*b^2 ne 0]; > e := [: d in disc]; > e := [: d in disc]; > e:=Sort(e); > [e[i] : i in [1..20]]; [ <17, -11, 6>, <27, 0, 16>, <32, -16, 0>, <32, -11, -14>, <32, -11, 14>, <32, -1, 0>, <32, 4, 0>, <36, -15, 22>, <36, 0, -27>, <36, 0, 1>, <37, -16, 16>, <40, -7, -6>, <40, -2, 1>, <40, 13, -34>, <52, -4, -3>, <52, 1, -10>, <53, 5, 22>, <54, 21, -26>, <55, 13, 14>, <56, -19, 30> ] > Conductor(EllipticCurve([0,0,1,-1,0])); 37 > disc:=[[a,b]: a in [-99..99], b in [-99..99] | 4*a^3+27*b^2 ne 0]; > e := [: d in disc]; > e:=Sort(e); > [e[i] : i in [1..20]]; [ <17, -11, 6>, <27, 0, 16>, <32, -81, 0>, <32, -16, 0>, <32, -11, -14>, <32, -11, 14>, <32, -1, 0>, <32, 4, 0>, <32, 64, 0>, <36, -15, 22>, <36, 0, -27>, <36, 0, 1>, <36, 0, 64>, <37, -16, 16>, <40, -32, 64>, <40, -7, -6>, <40, -2, 1>, <40, 13, -34>, <49, -35, -98>, <52, -4, -3> ] > dis:=[[a,b] : a in [-50..50], b in [-50..50] | 4*a^3+27*b^2 ne 0]; > e := Sort([ : d in dis]); > [e[i] : i in [1..10]]; [ <17, -11, 6>, <27, 0, 16>, <32, -16, 0>, <32, -11, -14>, <32, -11, 14>, <32, -1, 0>, <32, 4, 0>, <36, -15, 22>, <36, 0, -27>, <36, 0, 1> ] > [e[i] : i in [1..20]]; [ <17, -11, 6>, <27, 0, 16>, <32, -16, 0>, <32, -11, -14>, <32, -11, 14>, <32, -1, 0>, <32, 4, 0>, <36, -15, 22>, <36, 0, -27>, <36, 0, 1>, <37, -16, 16>, <40, -7, -6>, <40, -2, 1>, <40, 13, -34>, <52, -4, -3>, <52, 1, -10>, <53, 5, 22>, <54, 21, -26>, <55, 13, 14>, <56, -19, 30> ] > [e[i] : i in [1..40]]; [ <17, -11, 6>, <27, 0, 16>, <32, -16, 0>, <32, -11, -14>, <32, -11, 14>, <32, -1, 0>, <32, 4, 0>, <36, -15, 22>, <36, 0, -27>, <36, 0, 1>, <37, -16, 16>, <40, -7, -6>, <40, -2, 1>, <40, 13, -34>, <52, -4, -3>, <52, 1, -10>, <53, 5, 22>, <54, 21, -26>, <55, 13, 14>, <56, -19, 30>, <56, 1, 2>, <58, -19, 46>, <64, -4, 0>, <64, 1, 0>, <64, 16, 0>, <72, 6, -7>, <73, -19, -18>, <77, 32, 16>, <80, -7, 6>, <80, -2, -1>, <80, 13, 34>, <88, -4, 4>, <91, 16, 16>, <92, -1, 1>, <94, 5, -42>, <99, -27, -10>, <108, 0, 4>, <112, -19, -30>, <112, 1, -2>, <124, -17, -27> ] > [e[i] : i in [1..60]]; [ <17, -11, 6>, <27, 0, 16>, <32, -16, 0>, <32, -11, -14>, <32, -11, 14>, <32, -1, 0>, <32, 4, 0>, <36, -15, 22>, <36, 0, -27>, <36, 0, 1>, <37, -16, 16>, <40, -7, -6>, <40, -2, 1>, <40, 13, -34>, <52, -4, -3>, <52, 1, -10>, <53, 5, 22>, <54, 21, -26>, <55, 13, 14>, <56, -19, 30>, <56, 1, 2>, <58, -19, 46>, <64, -4, 0>, <64, 1, 0>, <64, 16, 0>, <72, 6, -7>, <73, -19, -18>, <77, 32, 16>, <80, -7, 6>, <80, -2, -1>, <80, 13, 34>, <88, -4, 4>, <91, 16, 16>, <92, -1, 1>, <94, 5, -42>, <99, -27, -10>, <108, 0, 4>, <112, -19, -30>, <112, 1, -2>, <124, -17, -27>, <144, -15, -22>, <144, 0, -1>, <144, 0, 27>, <144, 6, 7>, <162, 45, -18>, <176, -4, -4>, <179, -16, -48>, <180, -12, -11>, <184, 5, 6>, <189, -48, 16>, <200, 5, -10>, <208, -4, 3>, <208, 1, 10>, <216, -12, 20>, <216, -3, -34>, <238, 29, 30>, <243, 0, -48>, <244, 1, 6>, <248, 1, -1>, <256, -32, 0> ] > dis:=[[a,b] : a in [-100..100], b in [-100..100] | 4*a^3+27*b^2 ne 0]; > e := Sort([ : d in dis]); > Set([e[i][1] : i in [1..60]]); { 17, 27, 32, 36, 37, 40, 49, 52, 53, 54, 55, 56, 58, 62, 64, 72, 73, 77, 80, 88, 90, 91, 92, 94, 99, 108, 112, 124, 144, 162 } > M:=ModularSymbols(35); Creating M_2(Gamma_1(35),eps;F_0) , 0.111 seconds. > Print(Decomposition(M)); Parent at level 35 is creating child of level 7 Creating M_2(Gamma_1(7),eps;F_0) , 0.019 seconds. Parent at level 35 is creating child of level 5 Creating M_2(Gamma_1(5),eps;F_0) , 0.009 seconds. Sorting and labeling factors at level 35. Modular symbols factors: 35k2A: dim = 2 cuspidal 35k2B: dim = 4 cuspidal > E:=EllipticCurve(Decomposition(M)[1]); c4 = -418.172252902664551820539562557865 + 6.4623485337 E-27*i c6 = 1495.62296593433932404600259924260 - 3.8774091210 E-25*i Searching... Recomputing period integrals to higher precision c4 = -416.0000000057073550300467520047 + 1.2924697068 E-26*i c6 = 1447.99999989827071841138677334035 - 5.9453606512 E-25*i Candidate curve [-416,1448]: By Shimura-Taniyama, this is in the isogeny class. > Weierstrass(E); [ 11232, -78192 ] > EE:=EllipticCurve([-416,1448]); > Conductor(EE); 231354176 > EE:=EllipticCurve([11232,-78192]); > Conductor(EE); 35 > Em:=Decomposition(M)[1]; > RealTamagawa(Em); 1 > Print(Decomposition(ModularSymbols(43))); Creating M_2(Gamma_1(43),eps;F_0) , 0.079 seconds. Parent at level 43 is creating child of level 1 Creating M_2(Gamma_1(1),eps;F_0) Sorting and labeling factors at level 43. Modular symbols factors: 43k2A: dim = 2 cuspidal 43k2B: dim = 4 cuspidal 43k2C: dim = 1 eisenstein > C:=Decomposition(ModularSymbols(43))[2]; Creating M_2(Gamma_1(43),eps;F_0) , 0.08 seconds. Parent at level 43 is creating child of level 1 Creating M_2(Gamma_1(1),eps;F_0) Sorting and labeling factors at level 43. > TorsionBound(C); 7 [ 196, 196, 9604, 25921, 25921, 25921, 226576 ] > CuspOrder(C); 7 > TamagawaNumber(C); 7 > TamagawaNumber; Intrinsic 'TamagawaNumber' Signatures: ( E, p) -> RngIntElt The Tamagawa number of E at p. E must be defined over Q ( A) -> RngIntElt Compute the order of the group of Fp rational points of the component group of A at the largest prime which exactly divides the level of A. WARNING: Stein has not yet nailed down the power of 2 when Wp=+1! ( A, p) -> RngIntElt > LRatio(C); 2/7 > for x in [-50..50] do for y in [-50..50] do if y + y^2 - x^3 - x^2 eq 0 then\ print x," ",y; end if ; end for; end for; -1 -1 -1 0 0 -1 0 0 1 -2 1 1 2 -4 2 3 > M:=ModularSymbols(39); Creating M_2(Gamma_1(39),eps;F_0) , 0.109 seconds. > Print(Decomposition(M)); Parent at level 39 is creating child of level 13 Creating M_2(Gamma_1(13),eps;F_0) , 0.03 seconds. Parent at level 39 is creating child of level 3 Creating M_2(Gamma_1(3),eps;F_0) , 0.009 seconds. Sorting and labeling factors at level 39. Modular symbols factors: 39k2A: dim = 2 cuspidal 39k2B: dim = 4 cuspidal > for x in [-50..50] do for y in [-50..50] do if y + y^2 - x^3 - x^2-x eq 0 th\ en print x," ",y; end if ; end for; end for; 0 -1 0 0 > [e[i] : i in [1..60]]; [ <17, -11, 6>, <27, 0, 16>, <32, -81, 0>, <32, -16, 0>, <32, -11, -14>, <32, -11, 14>, <32, -1, 0>, <32, 4, 0>, <32, 64, 0>, <36, -15, 22>, <36, 0, -27>, <36, 0, 1>, <36, 0, 64>, <37, -16, 16>, <40, -32, 64>, <40, -7, -6>, <40, -2, 1>, <40, 13, -34>, <49, -35, -98>, <52, -4, -3>, <52, 1, -10>, <53, 5, 22>, <54, 21, -26>, <55, 13, 14>, <56, -19, 30>, <56, 1, 2>, <58, -19, 46>, <62, -11, 70>, <64, -64, 0>, <64, -4, 0>, <64, 1, 0>, <64, 16, 0>, <64, 81, 0>, <72, -39, -70>, <72, 6, -7>, <73, -19, -18>, <77, 32, 16>, <80, -32, -64>, <80, -7, 6>, <80, -2, -1>, <80, 13, 34>, <88, -4, 4>, <90, 93, 94>, <91, 16, 16>, <92, -16, 64>, <92, -1, 1>, <94, 5, -42>, <99, -27, -10>, <108, 0, 4>, <112, -19, -30>, <112, 1, -2>, <124, -17, -27>, <144, -39, 70>, <144, -15, -22>, <144, 0, -64>, <144, 0, -1>, <144, 0, 27>, <144, 6, 7>, <162, 45, -18>, <162, 69, 22> ] > D:=Decomposition(M); > E:=EllipticCurve(D[1]); c4 = 211.90336426093106978391115258196 + 0.E-28*i c6 = 2615.81027188545993519138996618140 + 0.E-27*i Searching... Recomputing period integrals to higher precision c4 = 216.99999895146677792213624697330 + 0.E-28*i c6 = 2754.99997827250759047615581648260 + 0.E-27*i Candidate curve [217,2755]: By Shimura-Taniyama, this is in the isogeny class. > Em:=D[1]; > E; Elliptic Curve defined by y^2 = x^3 - 5859*x - 148770 over Rational Field > RealVolume(E); >> RealVolume(E); ^ Runtime error in 'RealVolume': Bad argument types Argument types given: CurveEll > RealTamagawa; Intrinsic 'RealTamagawa' Signatures: ( A) -> RngIntElt Computes the number of real components of the abelian variety associated to A. > RealVolume; Intrinsic 'RealVolume' Signatures: ( A) -> FldPrElt ( A, n) -> FldPrElt Computes the volume of A(R), R the RealField. This function returns the volume of the identity component times the number RealTamagawa(A) of real components. > E; Elliptic Curve defined by y^2 = x^3 - 5859*x - 148770 over Rational Field > Em; Vector space of degree 9, dimension 2 over Rational Field User basis: ( 1 0 0 0 -2 0 0 2 1) ( 0 1 0 0 0 -1 0 0 0) > EE:=EllipticCurve([1,1,0,1,0]); > Weierstrass(EE); [ 621, -12690 ] > RealTamagawa(E); >> RealTamagawa(E); ^ Runtime error in 'RealTamagawa': Bad argument types Argument types given: CurveEll > RealTamagawa(Em); 2 > LAnalytic(Em); >> LAnalytic(Em); ^ Runtime error in 'LAnalytic': Bad argument types Argument types given: ModTupFld > LAnalytic(Em,1); (Using at least 67 terms of q-expansions.) 0.82668785170578791907065392744 > $1/6.6; 0.125255735106937563495553625369 > for x in [-50..50] do for y in [-50..50] do if y + y^2 - x^3 - x^2-x eq 0 th\ en print x," ",y; end if ; end for; end for; 0 -1 0 0 > MordelWeilGroup(E); >> MordelWeilGroup(E); ^ User error: Identifier 'MordelWeilGroup' has not been declared or assigned > MordelWeil(E); >> MordelWeil(E); ^ User error: Identifier 'MordelWeil' has not been declared or assigned > MordellWeil(E); >> MordellWeil(E); ^ User error: Identifier 'MordellWeil' has not been declared or assigned > MordellWeilGroup(E); Abelian Group isomorphic to Z/2 + Z/2 Defined on 2 generators Relations: 2*$.1 = 0 2*$.2 = 0 Mapping from: Abelian Group isomorphic to Z/2 + Z/2 Defined on 2 generators Relations: 2*$.1 = 0 2*$.2 = 0 to CurveEll: E > f,G:=MordellWeilGroup(E); > f(G.1); >> f(G.1); ^ Runtime error in '.': Bad argument types Argument types given: Map, RngIntElt > G,f:=MordellWeilGroup(E); > f(G.1); (87, 0, 1) > E; Elliptic Curve defined by y^2 = x^3 - 5859*x - 148770 over Rational Field > E:=MinimalModel(E); > f(G.1); (87, 0, 1) > f,G:=MordellWeilGroup(E); > f(G.1); >> f(G.1); ^ Runtime error in '.': Bad argument types Argument types given: Map, RngIntElt > G; Mapping from: GrpAb: f to CurveEll: E > G,f:=MordellWeilGroup(E); > f(G.1); (2, -1, 1) > Conductor(E); 39 > G; Abelian Group isomorphic to Z/2 + Z/2 Defined on 2 generators Relations: 2*G.1 = 0 2*G.2 = 0 > E; Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 4*x - 5 over Rational Field > E:=EllipticCurve([1,1,0,1,0]); > Conductor(E); 39 > G,f:=MordellWeilGroup(E); > f(G.1); (0, 0, 1) > f(2*G.1); (0, 1, 0) > E; Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + x over Rational Field > [e[i] : i in [1..60]]; [ <17, -11, 6>, <27, 0, 16>, <32, -81, 0>, <32, -16, 0>, <32, -11, -14>, <32, -11, 14>, <32, -1, 0>, <32, 4, 0>, <32, 64, 0>, <36, -15, 22>, <36, 0, -27>, <36, 0, 1>, <36, 0, 64>, <37, -16, 16>, <40, -32, 64>, <40, -7, -6>, <40, -2, 1>, <40, 13, -34>, <49, -35, -98>, <52, -4, -3>, <52, 1, -10>, <53, 5, 22>, <54, 21, -26>, <55, 13, 14>, <56, -19, 30>, <56, 1, 2>, <58, -19, 46>, <62, -11, 70>, <64, -64, 0>, <64, -4, 0>, <64, 1, 0>, <64, 16, 0>, <64, 81, 0>, <72, -39, -70>, <72, 6, -7>, <73, -19, -18>, <77, 32, 16>, <80, -32, -64>, <80, -7, 6>, <80, -2, -1>, <80, 13, 34>, <88, -4, 4>, <90, 93, 94>, <91, 16, 16>, <92, -16, 64>, <92, -1, 1>, <94, 5, -42>, <99, -27, -10>, <108, 0, 4>, <112, -19, -30>, <112, 1, -2>, <124, -17, -27>, <144, -39, 70>, <144, -15, -22>, <144, 0, -64>, <144, 0, -1>, <144, 0, 27>, <144, 6, 7>, <162, 45, -18>, <162, 69, 22> ] > E:=EllipticCurve([0,16]); > Conductor(E); 27 > TorsionSubgroup(E); Abelian Group isomorphic to Z/3 Defined on 1 generator Relations: 3*$.1 = 0 Mapping from: Abelian Group isomorphic to Z/3 Defined on 1 generator Relations: 3*$.1 = 0 to CurveEll: E > TamagawaNumber(E,3); 1 > E:=EllipticCurve([0,0,1,0,-7]); > f,G:=MordellWeilGroup(E); > WeierStrass(E); >> WeierStrass(E); ^ User error: Identifier 'WeierStrass' has not been declared or assigned > Weierstrass(E); [ 0, -314928 ] > E2:=EllipticCurve($1); > Conductor(E2); 27 > TorsionSubgroup(E2); Abelian Group isomorphic to Z/3 Defined on 1 generator Relations: 3*$.1 = 0 Mapping from: Abelian Group isomorphic to Z/3 Defined on 1 generator Relations: 3*$.1 = 0 to CurveEll: E2 > f,G:=MordellWeilGroup(E); > f,G:=MordellWeilGroup(E2); > f(G.1); >> f(G.1); ^ Runtime error in '.': Bad argument types Argument types given: Map, RngIntElt > G,f:=MordellWeilGroup(E2); > f(G.1); (108, 972, 1) > E; Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field > MinimalModel(E2); Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field Elliptic curve isomorphism from: CurveEll: E2 to CurveEll: E Taking (x, y, 1) to (1/36*x, 1/216*y - 1/2, 1) Elliptic curve isomorphism from: CurveEll: E to CurveEll: E2 Taking (x, y, 1) to (36*x, 216*y + 108, 1) > f(0*G.1); (0, 1, 0) > G,f:=MordellWeilGroup(E); > f(0*G.1); (0, 1, 0) > f(G.1); (3, 4, 1) > f(2*G.1); (3, -5, 1) > f(*G.1); >> f(*G.1); ^ User error: bad syntax > f(3*G.1); (0, 1, 0) > Em:=ModularFactor("27A"); Creating M_2(Gamma_1(27),eps;F_0) , 0.07 seconds. Parent at level 27 is creating child of level 9 Creating M_2(Gamma_1(9),eps;F_0) , 0.019 seconds. Sorting and labeling factors at level 27. > RealTamagawa(Em); 1 > for x in [-50..50] do for y in [-50..50] do if y + y^2 = x^3-7 eq 0 then pri\ nt x," ",y; end if ; end for; end for; >> for x in [-50..50] do for y in [-50..50] do if y + y^2 = x^3-7 eq 0 then pr ^ Runtime error in elt< ... >: LHS and RHS of relation constructor are not compatible > for x in [-50..50] do for y in [-50..50] do if y + y^2 -( x^3-7) eq 0 then p\ rint x," ",y; end if ; end for; end for; 3 -5 3 4 > for x in [-100..100] do for y in [-100..100] do if y + y^2 -( x^3-7) eq 0 th\ en print x," ",y; end if ; end for; end for; 3 -5 3 4 > LRatio(Em); 1/3 > Weierstrass(E); [ 0, -314928 ] > E; Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field > 314928 mod 3; 0 > MinimalModel(EllipticCurve([0,-314928])); Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field Elliptic curve isomorphism from: CurveEll: E2 to CurveEll: E Taking (x, y, 1) to (1/36*x, 1/216*y - 1/2, 1) Elliptic curve isomorphism from: CurveEll: E to CurveEll: E2 Taking (x, y, 1) to (36*x, 216*y + 108, 1) > for x in [0,4] do for y in [0..4] do if (y + y^2 -( x^3-7)) mod 5 eq 0 then \ print x," ",y; end if ; end for; end for; 4 1 4 3 > for x in [0..4] do for y in [0..4] do if (y + y^2 -( x^3-7)) mod 5 eq 0 then\ print x," ",y; end if ; end for; end for; 2 2 3 0 3 4 4 1 4 3 > p:=5;v:=[];for x in [0..p-1] do for y in [0..p-1] do if (y + y^2 -( x^3-7)) \ mod p eq 0 then print x," ",y; Append(~v,[x,y]); end if ; end for; end for; 2 2 3 0 3 4 4 1 4 3 > v; [ [ 2, 2 ], [ 3, 0 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ] ] > p:=2;v:=[];for x in [0..p-1] do for y in [0..p-1] do if (y + y^2 -( x^3-7)) \ mod p eq 0 then print x," ",y; Append(~v,[x,y]); end if ; end for; end for; 1 0 1 1 > v; [ [ 1, 0 ], [ 1, 1 ] ] > p:=7;v:=[];for x in [0..p-1] do for y in [0..p-1] do if (y + y^2 -( x^3-7)) \ mod p eq 0 then print x," ",y; Append(~v,[x,y]); end if ; end for; end for; 0 0 0 6 3 2 3 4 5 2 5 4 6 2 6 4 > #v; 8 > p:=11;v:=[];for x in [0..p-1] do for y in [0..p-1] do if (y + y^2 -( x^3-7))\ mod p eq 0 then print x," ",y; Append(~v,[x,y]); end if ; end for; end for; 2 3 2 7 3 4 3 6 4 1 4 9 5 5 6 0 6 10 7 2 7 8 > #v; 11 > p:=13;v:=[];for x in [0..p-1] do for y in [0..p-1] do if (y + y^2 -( x^3-7))\ mod p eq 0 then print x," ",y; Append(~v,[x,y]); end if ; end for; end for; 0 2 0 10 1 4 1 8 3 4 3 8 9 4 9 8 > #v; 8 > p:=17;v:=[];for x in [0..p-1] do for y in [0..p-1] do if (y + y^2 -( x^3-7))\ mod p eq 0 then print x," ",y; Append(~v,[x,y]); end if ; end for; end for; 3 4 3 12 4 2 4 14 6 7 6 9 7 5 7 11 8 3 8 13 9 6 9 10 12 8 14 0 14 16 15 1 15 15 > #v; 17 > p:=19;v:=[];for x in [0..p-1] do for y in [0..p-1] do if (y + y^2 -( x^3-7))\ mod p eq 0 then print x," ",y; Append(~v,[x,y]); end if ; end for; end for; 0 3 0 15 2 4 2 14 3 4 3 14 4 0 4 18 5 6 5 12 6 0 6 18 8 5 8 13 9 0 9 18 12 5 12 13 14 4 14 14 16 6 16 12 17 6 17 12 18 5 18 13 > #v; 26 > qEigenform(Em); q - 2*q^4 - q^7 > Print(Em); New cuspidal factor: dimension 2 (1), level 27, weight 2 (27A) > qEigenform(Em,20); q - 2*q^4 - q^7 + 5*q^13 + 4*q^16 - 7*q^19 > p:=13;v:=[];for x in [0..p-1] do for y in [0..p-1] do if (y + y^2 -( x^3-7))\ mod p eq 0 then print x," ",y; Append(~v,[x,y]); end if ; end for; end for; 0 2 0 10 1 4 1 8 3 4 3 8 9 4 9 8 > #v; 8 > RealVolume(Em); (Using at least 52 terms of q-expansions.) [1.76663863379948852756829393325 + 0.E-38*i] 1.76663863379948852756829393325 + 0.E-38*i > Print(Em); New cuspidal factor: dimension 2 (1), level 27, weight 2 (27A) > RealTamagawa(Em); 1 > LAnalytic(Em,1,200); 0.58887958342848331910456316656 > 0.58887/1.7666638; 0.333323182373465738076480652401 > A:=AffinePlane(Rationals()); > C:=AffineCurve(60*x^3+y^3+1); >> C:=AffineCurve(60*x^3+y^3+1); ^ Runtime error in 'AffineCurve': Bad argument types Argument types given: RngMPolElt > AffineCurve; Intrinsic 'AffineCurve' Signatures: ( D) -> CrvAffPl The affine model underlying D ( J) -> CrvAffPl The affine model underlying J > Curve > ; Intrinsic 'Curve' Signatures: ( J) -> CurveHyp The hyperelliptic curve from which the Jacobian J was obtained ( A, f) -> CrvAffPl The affine plane curve f = 0 ( P, f) -> CrvProjPl The projective plane curve f = 0 ( pl) -> Sch The curve of which pl is a place ( D) -> Sch The curve containing the elements of D ( d) -> Sch The curve on which d is a divisor ( J) -> Sch The curve containing the elements of J ( d) -> Sch The curve on which d is a jacobian point > C:=Curve(60*x^3+y^3+1); >> C:=Curve(60*x^3+y^3+1); ^ Runtime error in 'Curve': Bad argument types Argument types given: RngMPolElt > k:=Rationals(); > A:=AffineSpace(k,2); > C:=Curve(A,60*x^3+y^3+1); > C; Affine plane curve defined by 60*x^3 + y^3 + 1 > SingularPoints(C); [] > p:=A![0,-1]; > p in C; true > S,P,E:=WeierstrassForm(C,p); > E; Elliptic Curve defined by y^2 + 20*y = x^3 - 400/3 over Rational Field > Conductor(E); 24300 > mazur:=MinimalModel(E); > mazur; Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field > factor(24300); [ <2, 2>, <3, 5>, <5, 2> ] > Rank(E); 0 > TorsionSubgroup(E); Abelian Group of order 1 Mapping from: Abelian Group of order 1 to CurveEll: E > RealTamagawa(ModularFactor("11A")); Creating M_2(Gamma_1(11),eps;F_0) , 0.021 seconds. Parent at level 11 is creating child of level 1 Creating M_2(Gamma_1(1),eps;F_0) Sorting and labeling factors at level 11. 1 > DimensionSk(13,2); 0 > RealTamagawa(ModularFactor("14A")); Creating M_2(Gamma_1(14),eps;F_0) , 0.049 seconds. Parent at level 14 is creating child of level 7 Creating M_2(Gamma_1(7),eps;F_0) , 0.021 seconds. Parent at level 14 is creating child of level 2 Creating M_2(Gamma_1(2),eps;F_0) , 0.009 seconds. Sorting and labeling factors at level 14. 1 > DimensionSk(15,2); 1 > RealTamagawa(ModularFactor("15A")); Creating M_2(Gamma_1(15),eps;F_0) , 0.049 seconds. Parent at level 15 is creating child of level 5 Creating M_2(Gamma_1(5),eps;F_0) , 0.009 seconds. Parent at level 15 is creating child of level 3 Creating M_2(Gamma_1(3),eps;F_0) , 0.009 seconds. Sorting and labeling factors at level 15. 2 > RealTamagawa(ModularFactor("17A")); Creating M_2(Gamma_1(17),eps;F_0) , 0.03 seconds. Parent at level 17 is creating child of level 1 Creating M_2(Gamma_1(1),eps;F_0) Sorting and labeling factors at level 17. 1 > RealTamagawa(ModularFactor("19A")); Creating M_2(Gamma_1(19),eps;F_0) , 0.039 seconds. Parent at level 19 is creating child of level 1 Creating M_2(Gamma_1(1),eps;F_0) Sorting and labeling factors at level 19. 1 > RealTamagawa(ModularFactor("20A")); Creating M_2(Gamma_1(20),eps;F_0) , 0.069 seconds. Parent at level 20 is creating child of level 10 Creating M_2(Gamma_1(10),eps;F_0) , 0.03 seconds. Parent at level 20 is creating child of level 4 Creating M_2(Gamma_1(4),eps;F_0) , 0.009 seconds. Sorting and labeling factors at level 20. 1 > factor(24300); [ <2, 2>, <3, 5>, <5, 2> ] > A:=ModularFactor("65C"); Creating M_2(Gamma_1(65),eps;F_0) , 0.18 seconds. Parent at level 65 is creating child of level 13 Creating M_2(Gamma_1(13),eps;F_0) , 0.03 seconds. Parent at level 65 is creating child of level 5 Creating M_2(Gamma_1(5),eps;F_0) , 0.01 seconds. Sorting and labeling factors at level 65. > TamagawaNumber(A,5); 7 > TamagawaNumber(A,13); 1 > LRatio(A,1); 1/14 > quit; Total time: 84.859 seconds