> // J_0(43)
> M:=MS(43);
> C:=CS(M);
> D:=ND(C);
> D;
[
Modular symbols space of level 43, weight 2, and dimension 2,
Modular symbols space of level 43, weight 2, and dimension 4
]
> A:=D[2];
> ModularKernel(A);
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*$.1 = 0
2*$.2 = 0
> // However, however says "However, there are no endos of degree 4 that are totally positive."
> // However, with thought, however says, "However, I'm totally wrong!  2-sqrt(2)". 
> L1:=IntegralBasis(A); bL1:=[Representation(L1[i]) : i in [1..#L1]];
> bL1;
[
(0 1 0 0 0 0 0),
(0 0 0 1 0 0 0),
(0 0 0 0 0 1 0),
( 1  0 -1  0  1  0  0)
]
> Basis(A);
> Basis(A);
[
{-1/25, 0} + -1*{-1/17, 0} + {-1/14, 0},
{-1/39, 0},
{-1/32, 0},
{-1/21, 0}
]
> qEigenform(A,17);
q + a*q^2 - a*q^3 + (-a + 2)*q^5 - 2*q^6 + (a - 2)*q^7 - 2*a*q^8 - q^9 + (2*a - 2)*q^10 + (2*a - 1)*q^11 + (2*a + 1)*q^13 + (-2*a + 2)*q^14 + (-2*a + 2)*q^15 - 4*q^16 + O(q^17)
> Parent($1);
Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field
with modulus a^2 - 2
> Restrict(Tn(M,2),bL1);
[ 1  0  0 -1]
[ 0  1  1 -1]
[-1  1 -1  0]
[-1  0  0 -1]
> T2:=$1;
> S:=2-T2;
> MinimalPolynomial(S);
x^2 - 4*x + 2
> Roots(PolynomialRing(RealField())!x^2 - 4*x + 2);
[ <0.58578643762690495119831127579030192143, 1>, <3.4142135623730950488016887242096980785, 1> ]
> B:=D[1];
> IntersectionGroup(A,B);
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*$.1 = 0
2*$.2 = 0
> L1L2:=IntegralBasis(A+B);
> bL1L2:=[Representation(L1L2[i]) : i in [1..#L1L2]];
> L2:=IntegralBasis(B);
> bL1:=[Representation(L1[i]) : i in [1..#L1]];
>  bL2:=[Representation(L2[i]) : i in [1..#L2]];
> basis:=bL1 cat bL2;
> Coordinates(VectorSpaceWithBasis(basis),bL1L2[1]);
[ 1/2, 0, 0, 1/2, 1/2, 0 ]
> Coordinates(VectorSpaceWithBasis(basis),bL1L2[2]);
[ 1, 0, 0, 0, 0, 0 ]
> Coordinates(VectorSpaceWithBasis(basis),bL1L2[3]);
[ 0, 1/2, 1/2, -1/2, 0, 1/2 ]
> a:=VectorSpace(Q,4)![ 1/2, 0, 0, 1/2];
> b:=VectorSpace(Q,4)![ 0, 1/2, 1/2, -1/2];a:=VectorSpace(Q,4)![ 1/2, 0, 0, 1/2];
> a;
(1/2   0   0 1/2)
> b;
(   0  1/2  1/2 -1/2)
> a*S;
(1 0 0 2)
> b*S;
( 0  0  1 -1)
> LRatio(A,1);
2/7
> TamagawaNumber(A,43);
7
> TorsionBound(A,11);
7 [ 196, 196, 9604, 25921 ]
> Periods(A,97);
[
(0.5250281159132219433729491620 + 0.8066018577029307230283142376*i -0.2259499583067642118739519173 - 1.766644676299599532273333141*i),
(0.8241062742261960348649172089 - 0.1534409622571770568748354978*i 0.5981563162241222986475767259 - 1.920085638612119493276485634*i),
(-0.2990781583129740914919680469 - 0.9600428199601077799031497354*i -0.8241062745308865105215286433 - 0.1534409623125199610031524930*i),
(-0.8241062742261960348649172090 - 0.1534409622571770568748354978*i -0.5981563162241222986475767260 - 1.920085638612119493276485634*i)
]
> IntegralBasis(A);
[
{-1/39, 0},
{-1/32, 0},
{-1/21, 0},
{-1/25, 0} + -1*{-1/17, 0} + {-1/14, 0}
]
> 20^7*8;
10240000000
> 10^7*8;
80000000
> 20^6;
64000000
> C:=HyperellipticCurve(x^6+2*x+1);
> C;
Hyperelliptic Curve defined by y^2 = x^6 + 2*x + 1 over Rational Field
> J:=Jacobian(C);
> EulerFactor(J,7);
49*T^4 + 7*T^3 + 9*T^2 + T + 1
> EulerFactor(J,97);
9409*T^4 + 582*T^3 + 26*T^2 + 6*T + 1
> time C:=HyperellipticCurve(x^6+12*x+191); time J:=Jacobian(C); time [EulerFactor(J,p) : p in [11,17,37,97]];
Time: 0.009
Time: 0.000
[
121*T^4 + 77*T^3 + 28*T^2 + 7*T + 1,
289*T^4 - 17*T^3 + 16*T^2 - T + 1,
1369*T^4 - 74*T^3 + 46*T^2 - 2*T + 1,
9409*T^4 - 1649*T^3 + 187*T^2 - 17*T + 1
]
Time: 0.039
> 0.1*20^6;
6399999.999999999999999999999
> $1/(3600);
1777.777777777777777777777777
> 1777/24.0;
74.04166666666666666666666666
> save "showing.session";
Saving Magma state to "showing.session"
> quit;