torbound - 2011-10-01b
system:sage


{{{id=1|
attach "/home/wstein/Dropbox/current/torbound/code.sage"
///
}}}

{{{id=33|
A = EllipticCurve('15a')
B = EllipticCurve('30a')
///
}}}

{{{id=43|
A.torsion_subgroup().invariants()
///
(2, 4)
}}}

{{{id=45|
for E in A.isogeny_class()[0]:
    print E.torsion_subgroup().invariants()
///
(2, 4)
(2, 4)
(8,)
(2, 2)
(4,)
(4,)
(2,)
(2,)
}}}

{{{id=46|
for E in B.isogeny_class()[0]:
    print E.torsion_subgroup().invariants()
///
(6,)
(2, 6)
(2,)
(6,)
(6,)
(2, 2)
(2,)
(2,)
}}}

{{{id=44|
B.torsion_subgroup().invariants()
///
(6,)
}}}

{{{id=42|

///
}}}

{{{id=41|
A.isogeny_class()[1]
///
[ 1  2  2  2  4  4  4  4]
[ 2  1  4  4  2  2  8  8]
[ 2  4  1  4  8  8  8  8]
[ 2  4  4  1  8  8  2  2]
[ 4  2  8  8  1  4 16 16]
[ 4  2  8  8  4  1 16 16]
[ 4  8  8  2 16 16  1  4]
[ 4  8  8  2 16 16  4  1]
}}}

{{{id=34|
B.modular_degree()
///
2
}}}

{{{id=32|
J = J0(30)
D = J.decomposition(); D
///
[
Simple abelian subvariety 15a(1,30) of dimension 1 of J0(30),
Simple abelian subvariety 15a(2,30) of dimension 1 of J0(30),
Simple abelian subvariety 30a(1,30) of dimension 1 of J0(30)
]
}}}

{{{id=31|
D[2].intersection(D[0] + D[1])
///
(Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 30a(1,30) of dimension 1 of J0(30), Simple abelian subvariety of dimension 0 of J0(30))
}}}

{{{id=30|
D[0].intersection(D[1])
///
(Finite subgroup with invariants [4] over QQ of Simple abelian subvariety 15a(1,30) of dimension 1 of J0(30), Simple abelian subvariety of dimension 0 of J0(30))
}}}

{{{id=39|
T = ModTor(Gamma0(30)); T
///
got M 0.0
got S 0.0
got S_Z 0.0
Torsion on the modular Jacobian associated to Congruence Subgroup Gamma0(30)
}}}

{{{id=38|
E7 = T.E_lattice([7]); E7
///
Free module of degree 6 and rank 6 over Integer Ring
Echelon basis matrix:
[1/48 1/12 1/48    0  7/8 5/48]
[   0  1/8    0  3/8  1/8    0]
[   0    0    1    0    0    0]
[   0    0    0  1/2    0    0]
[   0    0    0    0    1    0]
[   0    0    0    0    0  1/2]
}}}

{{{id=40|
E7 / ZZ^6
///
Finitely generated module V/W over Integer Ring with invariants (2, 2, 8, 48)
}}}

{{{id=37|

///
}}}

{{{id=36|

///
}}}

{{{id=29|

///
}}}

{{{id=2|
M = ModTor(Gamma0(24))
///
got M 0.0
got S 0.0
got S_Z 0.0
}}}

{{{id=3|
C = M.C_lattice(); C
///
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/4   0]
[  0 1/2]
}}}

{{{id=35|

///
}}}

{{{id=5|
C.basis_matrix().det()^(-1)
///
8
}}}

{{{id=11|
E0 = M.E_lattice(prime_range(5,200), False); E0
///
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/8   0]
[  0 1/8]
}}}

{{{id=6|
E = M.E_lattice(prime_range(5,200)); E
///
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/8   0]
[  0 1/2]
}}}

{{{id=7|
M.M
///
Modular Symbols space of dimension 9 for Gamma_0(24) of weight 2 with sign 0 over Rational Field
}}}

{{{id=8|
M.S
///
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(24) of weight 2 with sign 0 over Rational Field
}}}

{{{id=9|
J = EllipticCurve('24a')
J.torsion_order()
///
8
}}}

{{{id=22|
J.ap(2) - (2+1)
///
-3
}}}

{{{id=23|
J.ap(3) - (3+1)
///
-5
}}}

{{{id=24|
J.discriminant().factor()
///
2^8 * 3^2
}}}

{{{id=28|

///
}}}

{{{id=27|

///
}}}

{{{id=26|

///
}}}

{{{id=25|

///
}}}

{{{id=13|
J.ap(5)
///
-2
}}}

{{{id=10|
latex(EllipticCurve('24a'))
///
y^2 = x^3 - x^2 - 4x + 4 
}}}

{{{id=12|
J.plot()
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=14|
R.<x> = RR[]
(x^3 - x^2 - 4*x + 4).roots()
///
[(-2.00000000000000, 1), (1.00000000000000, 1), (2.00000000000000, 1)]
}}}

{{{id=15|
J.torsion_subgroup()
///
Torsion Subgroup isomorphic to Z/2 + Z/4 associated to the Elliptic Curve defined by y^2 = x^3 - x^2 - 4*x + 4 over Rational Field
}}}

{{{id=16|
J = J0(24)
///
}}}

{{{id=17|
J.rational_cuspidal_subgroup()
///
Finite subgroup with invariants [2, 4] over QQ of Abelian variety J0(24) of dimension 1
}}}

{{{id=21|

///
}}}

{{{id=20|

///
}}}

{{{id=19|

///
}}}

{{{id=18|

///
}}}