Base class for modular abelian varieties¶

AUTHORS:

William Stein (2007-03)

class sage.modular.abvar.abvar.ModularAbelianVariety(groups, lattice=None, base_field=Rational Field, is_simple=None, newform_level=None, isogeny_number=None, number=None, check=True)¶

Bases: sage.modular.abvar.abvar.ModularAbelianVariety_abstract

Create a modular abelian variety with given level and base field.

INPUT:

groups - a tuple of congruence subgroups
lattice - (default: \(\ZZ^n\)) a full lattice in \(\ZZ^n\), where \(n\) is the sum of the dimensions of the spaces of cuspidal modular symbols corresponding to each \(\Gamma \in\) groups
base_field - a field (default: \(\QQ\))

EXAMPLES:

sage: J0(23)
Abelian variety J0(23) of dimension 2

lattice()¶

Return the lattice that defines this abelian variety.

OUTPUT:

lattice - a lattice embedded in the rational homology of the ambient product Jacobian

EXAMPLES:

sage: A = (J0(11) * J0(37))[1]; A
Simple abelian subvariety 37a(1,37) of dimension 1 of J0(11) x J0(37)
sage: type(A)
<class 'sage.modular.abvar.abvar.ModularAbelianVariety_with_category'>
sage: A.lattice()
Free module of degree 6 and rank 2 over Integer Ring
Echelon basis matrix:
[ 0  0  1 -1  1  0]
[ 0  0  0  0  2 -1]

class sage.modular.abvar.abvar.ModularAbelianVariety_abstract(groups, base_field, is_simple=None, newform_level=None, isogeny_number=None, number=None, check=True)¶

Bases: sage.structure.parent.Parent

Abstract base class for modular abelian varieties.

INPUT:

groups - a tuple of congruence subgroups
base_field - a field
is_simple - bool; whether or not self is simple
newform_level - if self is isogenous to a newform abelian variety, returns the level of that abelian variety
isogeny_number - which isogeny class the corresponding newform is in; this corresponds to the Cremona letter code
number - the t number of the degeneracy map that this abelian variety is the image under
check - whether to do some type checking on the defining data

EXAMPLES: One should not create an instance of this class, but we do so anyways here as an example:

sage: A = sage.modular.abvar.abvar.ModularAbelianVariety_abstract((Gamma0(37),), QQ)
sage: type(A)
<class 'sage.modular.abvar.abvar.ModularAbelianVariety_abstract_with_category'>

All hell breaks loose if you try to do anything with \(A\):

sage: A
<repr(<sage.modular.abvar.abvar.ModularAbelianVariety_abstract_with_category at 0x...>) failed: NotImplementedError: BUG -- lattice method must be defined in derived class>

All instances of this class are in the category of modular abelian varieties:

sage: A.category()
Category of modular abelian varieties over Rational Field
sage: J0(23).category()
Category of modular abelian varieties over Rational Field

ambient_morphism()¶

Return the morphism from self to the ambient variety. This is injective if self is natural a subvariety of the ambient product Jacobian.

OUTPUT: morphism

The output is cached.

EXAMPLES: We compute the ambient structure morphism for an abelian subvariety of \(J_0(33)\):

sage: A,B,C = J0(33)
sage: phi = A.ambient_morphism()
sage: phi.domain()
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: phi.codomain()
Abelian variety J0(33) of dimension 3
sage: phi.matrix()
[ 1  1 -2  0  2 -1]
[ 0  3 -2 -1  2  0]

phi is of course injective

sage: phi.kernel()
(Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33),
 Abelian subvariety of dimension 0 of J0(33))

This is the same as the basis matrix for the lattice corresponding to self:

sage: A.lattice()
Free module of degree 6 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1  1 -2  0  2 -1]
[ 0  3 -2 -1  2  0]

We compute a non-injective map to an ambient space:

sage: Q,pi = J0(33)/A
sage: phi = Q.ambient_morphism()
sage: phi.matrix()
[  1   4   1   9  -1  -1]
[  0  15   0   0  30 -75]
[  0   0   5  10  -5  15]
[  0   0   0  15 -15  30]
sage: phi.kernel()[0]
Finite subgroup with invariants [5, 15, 15] over QQ of Abelian variety factor of dimension 2 of J0(33)

ambient_variety()¶

Return the ambient modular abelian variety that contains this abelian variety. The ambient variety is always a product of Jacobians of modular curves.

OUTPUT: abelian variety

EXAMPLES:

sage: A = J0(33)[0]; A
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: A.ambient_variety()
Abelian variety J0(33) of dimension 3

base_extend(K)¶

EXAMPLES:

sage: A = J0(37); A
Abelian variety J0(37) of dimension 2
sage: A.base_extend(QQbar)
Abelian variety J0(37) over Algebraic Field of dimension 2
sage: A.base_extend(GF(7))
Abelian variety J0(37) over Finite Field of size 7 of dimension 2

base_field()¶

Synonym for self.base_ring().

EXAMPLES:

sage: J0(11).base_field()
Rational Field

change_ring(R)¶

Change the base ring of this modular abelian variety.

EXAMPLES:

sage: A = J0(23)
sage: A.change_ring(QQ)
Abelian variety J0(23) of dimension 2

complement(A=None)¶

Return a complement of this abelian variety.

INPUT:

A - (default: None); if given, A must be an abelian variety that contains self, in which case the complement of self is taken inside A. Otherwise the complement is taken in the ambient product Jacobian.

OUTPUT: abelian variety

EXAMPLES:

sage: a,b,c = J0(33)
sage: (a+b).complement()
Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
sage: (a+b).complement() == c
True
sage: a.complement(a+b)
Abelian subvariety of dimension 1 of J0(33)

conductor()¶

Return the conductor of this abelian variety.

EXAMPLES:

sage: A = J0(23)
sage: A.conductor().factor()
23^2

sage: A = J1(25)
sage: A.conductor().factor()
5^24

sage: A = J0(11^2); A.decomposition()
[
Simple abelian subvariety 11a(1,121) of dimension 1 of J0(121),
Simple abelian subvariety 11a(11,121) of dimension 1 of J0(121),
Simple abelian subvariety 121a(1,121) of dimension 1 of J0(121),
Simple abelian subvariety 121b(1,121) of dimension 1 of J0(121),
Simple abelian subvariety 121c(1,121) of dimension 1 of J0(121),
Simple abelian subvariety 121d(1,121) of dimension 1 of J0(121)
]
sage: A.conductor().factor()
11^10

sage: A = J0(33)[0]; A
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: A.conductor()
11
sage: A.elliptic_curve().conductor()
11

cuspidal_subgroup()¶

Return the cuspidal subgroup of this modular abelian variety. This is the subgroup generated by rational cusps.

EXAMPLES:

sage: J = J0(54)
sage: C = J.cuspidal_subgroup()
sage: C.gens()
[[(1/3, 0, 0, 0, 0, 1/3, 0, 2/3)], [(0, 1/3, 0, 0, 0, 2/3, 0, 1/3)], [(0, 0, 1/9, 1/9, 1/9, 1/9, 1/9, 2/9)], [(0, 0, 0, 1/3, 0, 1/3, 0, 0)], [(0, 0, 0, 0, 1/3, 1/3, 0, 1/3)], [(0, 0, 0, 0, 0, 0, 1/3, 2/3)]]
sage: C.invariants()
[3, 3, 3, 3, 3, 9]
sage: J1(13).cuspidal_subgroup()
Finite subgroup with invariants [19, 19] over QQ of Abelian variety J1(13) of dimension 2
sage: A = J0(33)[0]
sage: A.cuspidal_subgroup()
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)

decomposition(simple=True, bound=None)¶

Return a sequence of abelian subvarieties of self that are all simple, have finite intersection and sum to self.

INPUT: simple- bool (default: True) if True, all factors are simple. If False, each factor returned is isogenous to a power of a simple and the simples in each factor are distinct.

bound - int (default: None) if given, only use Hecke operators up to this bound when decomposing. This can give wrong answers, so use with caution!

EXAMPLES:

sage: m = ModularSymbols(11).cuspidal_submodule()
sage: d1 = m.degeneracy_map(33,1).matrix(); d3=m.degeneracy_map(33,3).matrix()
sage: w = ModularSymbols(33).submodule((d1 + d3).image(), check=False)
sage: A = w.abelian_variety(); A
Abelian subvariety of dimension 1 of J0(33)
sage: D = A.decomposition(); D
[
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33)
]
sage: D[0] == A
True
sage: B = A + J0(33)[0]; B
Abelian subvariety of dimension 2 of J0(33)
sage: dd = B.decomposition(simple=False); dd
[
Abelian subvariety of dimension 2 of J0(33)
]
sage: dd[0] == B
True
sage: dd = B.decomposition(); dd
[
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33),
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33)
]
sage: sum(dd) == B
True

We decompose a product of two Jacobians:

sage: (J0(33) * J0(11)).decomposition()
[
Simple abelian subvariety 11a(1,11) of dimension 1 of J0(33) x J0(11),
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) x J0(11),
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33) x J0(11),
Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) x J0(11)
]

degen_t(none_if_not_known=False)¶

If this abelian variety is obtained via decomposition then it gets labeled with the newform label along with some information about degeneracy maps. In particular, the label ends in a pair \((t,N)\), where \(N\) is the ambient level and \(t\) is an integer that divides the quotient of \(N\) by the newform level. This function returns the tuple \((t,N)\), or raises a ValueError if self isn’t simple.

Note

It need not be the case that self is literally equal to the image of the newform abelian variety under the \(t^{th}\) degeneracy map. See the documentation for the label method for more details.

INPUT:

none_if_not_known - (default: False) - if True, return None instead of attempting to compute the degen map’s \(t\), if it isn’t known. This None result is not cached.

OUTPUT: a pair (integer, integer)

EXAMPLES:

sage: D = J0(33).decomposition(); D
[
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33),
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33),
Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
]
sage: D[0].degen_t()
(1, 33)
sage: D[1].degen_t()
(3, 33)
sage: D[2].degen_t()
(1, 33)
sage: J0(33).degen_t()
Traceback (most recent call last):
...
ValueError: self must be simple

degeneracy_map(M_ls, t_ls)¶

Return the degeneracy map with domain self and given level/parameter. If self.ambient_variety() is a product of Jacobians (as opposed to a single Jacobian), then one can provide a list of new levels and parameters, corresponding to the ambient Jacobians in order. (See the examples below.)

INPUT:

M, t - integers level and \(t\), or
Mlist, tlist - if self is in a nontrivial product ambient Jacobian, input consists of a list of levels and corresponding list of \(t\)’s.

OUTPUT: a degeneracy map

EXAMPLES: We make several degeneracy maps related to \(J_0(11)\) and \(J_0(33)\) and compute their matrices.

sage: d1 = J0(11).degeneracy_map(33, 1); d1
Degeneracy map from Abelian variety J0(11) of dimension 1 to Abelian variety J0(33) of dimension 3 defined by [1]
sage: d1.matrix()
[ 0 -3  2  1 -2  0]
[ 1 -2  0  1  0 -1]
sage: d2 = J0(11).degeneracy_map(33, 3); d2
Degeneracy map from Abelian variety J0(11) of dimension 1 to Abelian variety J0(33) of dimension 3 defined by [3]
sage: d2.matrix()
[-1  0  0  0  1 -2]
[-1 -1  1 -1  1  0]
sage: d3 = J0(33).degeneracy_map(11, 1); d3
Degeneracy map from Abelian variety J0(33) of dimension 3 to Abelian variety J0(11) of dimension 1 defined by [1]

He we verify that first mapping from level \(11\) to level \(33\), then back is multiplication by \(4\):

sage: d1.matrix() * d3.matrix()
[4 0]
[0 4]

We compute a more complicated degeneracy map involving nontrivial product ambient Jacobians; note that this is just the block direct sum of the two matrices at the beginning of this example:

sage: d = (J0(11)*J0(11)).degeneracy_map([33,33], [1,3]); d
Degeneracy map from Abelian variety J0(11) x J0(11) of dimension 2 to Abelian variety J0(33) x J0(33) of dimension 6 defined by [1, 3]
sage: d.matrix()
[ 0 -3  2  1 -2  0  0  0  0  0  0  0]
[ 1 -2  0  1  0 -1  0  0  0  0  0  0]
[ 0  0  0  0  0  0 -1  0  0  0  1 -2]
[ 0  0  0  0  0  0 -1 -1  1 -1  1  0]

degree()¶

Return the degree of this abelian variety, which is the dimension of the ambient Jacobian product.

EXAMPLES:

sage: A = J0(23)
sage: A.dimension()
2

dimension()¶

Return the dimension of this abelian variety.

EXAMPLES:

sage: A = J0(23)
sage: A.dimension()
2

direct_product(other)¶

Compute the direct product of self and other.

INPUT:

self, other - modular abelian varieties

OUTPUT: abelian variety

EXAMPLES:

sage: J0(11).direct_product(J1(13))
Abelian variety J0(11) x J1(13) of dimension 3
sage: A = J0(33)[0].direct_product(J0(33)[1]); A
Abelian subvariety of dimension 2 of J0(33) x J0(33)
sage: A.lattice()
Free module of degree 12 and rank 4 over Integer Ring
Echelon basis matrix:
[ 1  1 -2  0  2 -1  0  0  0  0  0  0]
[ 0  3 -2 -1  2  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  1  0  0  0 -1  2]
[ 0  0  0  0  0  0  0  1 -1  1  0 -2]

dual()¶

Return the dual of this abelian variety.

OUTPUT:

dual abelian variety
morphism from self to dual
covering morphism from J to dual

Warning

This is currently only implemented when self is an abelian subvariety of the ambient Jacobian product, and the complement of self in the ambient product Jacobian share no common factors. A more general implementation will require implementing computation of the intersection pairing on integral homology and the resulting Weil pairing on torsion.

EXAMPLES: We compute the dual of the elliptic curve newform abelian variety of level \(33\), and find the kernel of the modular map, which has structure \((\ZZ/3)^2\).

sage: A,B,C = J0(33)
sage: C
Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
sage: Cd, f, pi = C.dual()
sage: f.matrix()
[3 0]
[0 3]
sage: f.kernel()[0]
Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)

By a theorem the modular degree must thus be \(3\):

sage: E = EllipticCurve('33a')
sage: E.modular_degree()
3

Next we compute the dual of a \(2\)-dimensional new simple abelian subvariety of \(J_0(43)\).

sage: A = AbelianVariety('43b'); A
Newform abelian subvariety 43b of dimension 2 of J0(43)
sage: Ad, f, pi = A.dual()

The kernel shows that the modular degree is \(2\):

sage: f.kernel()[0]
Finite subgroup with invariants [2, 2] over QQ of Newform abelian subvariety 43b of dimension 2 of J0(43)

Unfortunately, the dual is not implemented in general:

sage: A = J0(22)[0]; A
Simple abelian subvariety 11a(1,22) of dimension 1 of J0(22)
sage: A.dual()
Traceback (most recent call last):
...
NotImplementedError: dual not implemented unless complement shares no simple factors with self.

elliptic_curve()¶

Return an elliptic curve isogenous to self. If self is not dimension 1 with rational base ring, raise a ValueError.

The elliptic curve is found by looking it up in the CremonaDatabase. The CremonaDatabase contains all curves up to some large conductor. If a curve is not found in the CremonaDatabase, a RuntimeError will be raised. In practice, only the most committed users will see this RuntimeError.

OUTPUT: an elliptic curve isogenous to self.

EXAMPLES:

sage: J = J0(11)
sage: J.elliptic_curve()
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field

sage: J = J0(49)
sage: J.elliptic_curve()
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 2*x - 1 over Rational Field

sage: A = J0(37)[1]
sage: E = A.elliptic_curve()
sage: A.lseries()(1)
0.725681061936153
sage: E.lseries()(1)
0.725681061936153

Elliptic curves are of dimension 1.

sage: J = J0(23)
sage: J.elliptic_curve()
Traceback (most recent call last):
...
ValueError: self must be of dimension 1

This is only implemented for curves over QQ.

sage: J = J0(11).change_ring(CC)
sage: J.elliptic_curve()
Traceback (most recent call last):
...
ValueError: base ring must be QQ

endomorphism_ring(category=None)¶

Return the endomorphism ring of self.

OUTPUT: b = self.sturm_bound()

EXAMPLES: We compute a few endomorphism rings:

sage: J0(11).endomorphism_ring()
Endomorphism ring of Abelian variety J0(11) of dimension 1
sage: J0(37).endomorphism_ring()
Endomorphism ring of Abelian variety J0(37) of dimension 2
sage: J0(33)[2].endomorphism_ring()
Endomorphism ring of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)

No real computation is done:

sage: J1(123456).endomorphism_ring()
Endomorphism ring of Abelian variety J1(123456) of dimension 423185857

finite_subgroup(X, field_of_definition=None, check=True)¶

Return a finite subgroup of this modular abelian variety.

INPUT:

X - list of elements of other finite subgroups of this modular abelian variety or elements that coerce into the rational homology (viewed as a rational vector space); also X could be a finite subgroup itself that is contained in this abelian variety.
field_of_definition - (default: None) field over which this group is defined. If None try to figure out the best base field.

OUTPUT: a finite subgroup of a modular abelian variety

EXAMPLES:

sage: J = J0(11)
sage: J.finite_subgroup([[1/5,0], [0,1/3]])
Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1

sage: J = J0(33); C = J[0].cuspidal_subgroup(); C
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: J.finite_subgroup([[0,0,0,0,0,1/6]])
Finite subgroup with invariants [6] over QQbar of Abelian variety J0(33) of dimension 3
sage: J.finite_subgroup(C)
Finite subgroup with invariants [5] over QQ of Abelian variety J0(33) of dimension 3

This method gives a way of changing the ambient abelian variety of a finite subgroup. This caused an issue in trac ticket #6392 but should be fixed now.

sage: A, B = J0(43)
sage: G, _ = A.intersection(B)
sage: G
Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 43a(1,43) of dimension 1 of J0(43)
sage: B.finite_subgroup(G)
Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43)

free_module()¶

Synonym for self.lattice().

OUTPUT: a free module over \(\ZZ\)

EXAMPLES:

sage: J0(37).free_module()
Ambient free module of rank 4 over the principal ideal domain Integer Ring
sage: J0(37)[0].free_module()
Free module of degree 4 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1 -1  1  0]
[ 0  0  2 -1]

frobenius_polynomial(p, var='x')¶

Computes the frobenius polynomial at \(p\).

INPUT:

p – prime number

OUTPUT:

a monic integral polynomial

EXAMPLES:

sage: f = Newform('39b','a')
sage: A=AbelianVariety(f)
sage: A.frobenius_polynomial(5)
x^4 + 2*x^2 + 25

sage: J=J0(23)
sage: J.frobenius_polynomial(997)
x^4 + 20*x^3 + 1374*x^2 + 19940*x + 994009

sage: J = J0(33)
sage: J.frobenius_polynomial(7)
x^6 + 9*x^4 - 16*x^3 + 63*x^2 + 343

sage: J = J0(19)
sage: J.frobenius_polynomial(3, var='y')
y^2 + 2*y + 3

sage: J = J0(3); J
Abelian variety J0(3) of dimension 0
sage: J.frobenius_polynomial(11)
1

sage: A = J1(27)[1]; A
Simple abelian subvariety 27bG1(1,27) of dimension 12 of J1(27)
sage: A.frobenius_polynomial(11)
x^24 - 3*x^23 - 15*x^22 + 126*x^21 - 201*x^20 - 1488*x^19 + 7145*x^18 - 1530*x^17 - 61974*x^16 + 202716*x^15 - 19692*x^14 - 1304451*x^13 + 4526883*x^12 - 14348961*x^11 - 2382732*x^10 + 269814996*x^9 - 907361334*x^8 - 246408030*x^7 + 12657803345*x^6 - 28996910448*x^5 - 43086135081*x^4 + 297101409066*x^3 - 389061369015*x^2 - 855935011833*x + 3138428376721

sage: J = J1(33)
sage: J.frobenius_polynomial(11)
Traceback (most recent call last):
...
ValueError: p must not divide the level of self
sage: J.frobenius_polynomial(4)
Traceback (most recent call last):
...
ValueError: p must be prime

groups()¶

Return an ordered tuple of the congruence subgroups that the ambient product Jacobian is attached to.

Every modular abelian variety is a finite quotient of an abelian subvariety of a product of modular Jacobians \(J_\Gamma\). This function returns a tuple containing the groups \(\Gamma\).

EXAMPLES:

sage: A = (J0(37) * J1(13))[0]; A
Simple abelian subvariety 13aG1(1,13) of dimension 2 of J0(37) x J1(13)
sage: A.groups()
(Congruence Subgroup Gamma0(37), Congruence Subgroup Gamma1(13))

hecke_operator(n)¶

Return the \(n^{th}\) Hecke operator on the modular abelian variety, if this makes sense [[elaborate]]. Otherwise raise a ValueError.

EXAMPLES: We compute \(T_2\) on \(J_0(37)\).

sage: t2 = J0(37).hecke_operator(2); t2
Hecke operator T_2 on Abelian variety J0(37) of dimension 2
sage: t2.charpoly().factor()
x * (x + 2)
sage: t2.index()
2

Note that there is no matrix associated to Hecke operators on modular abelian varieties. For a matrix, instead consider, e.g., the Hecke operator on integral or rational homology.

sage: t2.action_on_homology().matrix()
[-1  1  1 -1]
[ 1 -1  1  0]
[ 0  0 -2  1]
[ 0  0  0  0]

hecke_polynomial(n, var='x')¶

Return the characteristic polynomial of the \(n^{th}\) Hecke operator \(T_n\) acting on self. Raises an ArithmeticError if self is not Hecke equivariant.

INPUT:

n - integer \(\geq 1\)
var - string (default: ‘x’); valid variable name

EXAMPLES:

sage: J0(33).hecke_polynomial(2)
x^3 + 3*x^2 - 4
sage: f = J0(33).hecke_polynomial(2, 'y'); f
y^3 + 3*y^2 - 4
sage: f.parent()
Univariate Polynomial Ring in y over Rational Field
sage: J0(33)[2].hecke_polynomial(3)
x + 1
sage: J0(33)[0].hecke_polynomial(5)
x - 1
sage: J0(33)[0].hecke_polynomial(11)
x - 1
sage: J0(33)[0].hecke_polynomial(3)
Traceback (most recent call last):
...
ArithmeticError: subspace is not invariant under matrix

homology(base_ring=Integer Ring)¶

Return the homology of this modular abelian variety.

Warning

For efficiency reasons the basis of the integral homology need not be the same as the basis for the rational homology.

EXAMPLES:

sage: J0(389).homology(GF(7))
Homology with coefficients in Finite Field of size 7 of Abelian variety J0(389) of dimension 32
sage: J0(389).homology(QQ)
Rational Homology of Abelian variety J0(389) of dimension 32
sage: J0(389).homology(ZZ)
Integral Homology of Abelian variety J0(389) of dimension 32

in_same_ambient_variety(other)¶

Return True if self and other are abelian subvarieties of the same ambient product Jacobian.

EXAMPLES:

sage: A,B,C = J0(33)
sage: A.in_same_ambient_variety(B)
True
sage: A.in_same_ambient_variety(J0(11))
False

integral_homology()¶

Return the integral homology of this modular abelian variety.

EXAMPLES:

sage: H = J0(43).integral_homology(); H
Integral Homology of Abelian variety J0(43) of dimension 3
sage: H.rank()
6
sage: H = J1(17).integral_homology(); H
Integral Homology of Abelian variety J1(17) of dimension 5
sage: H.rank()
10

If you just ask for the rank of the homology, no serious calculations are done, so the following is fast:

sage: H = J0(50000).integral_homology(); H
Integral Homology of Abelian variety J0(50000) of dimension 7351
sage: H.rank()
14702

A product:

sage: H = (J0(11) * J1(13)).integral_homology()
sage: H.hecke_operator(2)
Hecke operator T_2 on Integral Homology of Abelian variety J0(11) x J1(13) of dimension 3
sage: H.hecke_operator(2).matrix()
[-2  0  0  0  0  0]
[ 0 -2  0  0  0  0]
[ 0  0 -1 -1 -1  1]
[ 0  0  1 -2 -1  0]
[ 0  0  0  0 -2  1]
[ 0  0  0  0 -1 -1]

intersection(other)¶

Return the intersection of self and other inside a common ambient Jacobian product.

When other is a modular abelian variety, the output will be a tuple (G, A), where G is a finite subgroup that surjects onto the component group and A is the identity component. So in particular, the intersection is the variety G+A. Note that G is not chosen in any canonical way. When other is a finite group, the intersection will be returned as a finite group.

INPUT:

other - a modular abelian variety or a finite group

OUTPUT: If other is a modular abelian variety:

G - finite subgroup of self
A - abelian variety (identity component of intersection)

If other is a finite group:

G - a finite group

EXAMPLES: We intersect some abelian varieties with finite intersection.

sage: J = J0(37)
sage: J[0].intersection(J[1])
(Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37), Simple abelian subvariety of dimension 0 of J0(37))

sage: D = list(J0(65)); D
[Simple abelian subvariety 65a(1,65) of dimension 1 of J0(65), Simple abelian subvariety 65b(1,65) of dimension 2 of J0(65), Simple abelian subvariety 65c(1,65) of dimension 2 of J0(65)]
sage: D[0].intersection(D[1])
(Finite subgroup with invariants [2] over QQ of Simple abelian subvariety 65a(1,65) of dimension 1 of J0(65), Simple abelian subvariety of dimension 0 of J0(65))
sage: (D[0]+D[1]).intersection(D[1]+D[2])
(Finite subgroup with invariants [2] over QQbar of Abelian subvariety of dimension 3 of J0(65), Abelian subvariety of dimension 2 of J0(65))

sage: J = J0(33)
sage: J[0].intersection(J[1])
(Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Simple abelian subvariety of dimension 0 of J0(33))

Next we intersect two abelian varieties with non-finite intersection:

sage: J = J0(67); D = J.decomposition(); D
[
Simple abelian subvariety 67a(1,67) of dimension 1 of J0(67),
Simple abelian subvariety 67b(1,67) of dimension 2 of J0(67),
Simple abelian subvariety 67c(1,67) of dimension 2 of J0(67)
]
sage: (D[0] + D[1]).intersection(D[1] + D[2])
(Finite subgroup with invariants [5, 10] over QQbar of Abelian subvariety of dimension 3 of J0(67), Abelian subvariety of dimension 2 of J0(67))

When the intersection is infinite, the output is (G, A), where G surjects onto the component group. This choice of G is not canonical (see trac ticket #26189). In this following example, B is a subvariety of J:

sage: d1 = J0(11).degeneracy_map(22, 1)
sage: d2 = J0(11).degeneracy_map(22, 2)
sage: B = (d1-d2).image()
sage: J = J0(22)
sage: J.intersection(B)
(Finite subgroup with invariants [] over QQbar of Abelian variety J0(22) of dimension 2,
 Abelian subvariety of dimension 1 of J0(22))
sage: G, B = B.intersection(J); G, B
(Finite subgroup with invariants [2] over QQbar of Abelian subvariety of dimension 1 of J0(22),
 Abelian subvariety of dimension 1 of J0(22))
sage: G.is_subgroup(B)
True

is_J0()¶

Return whether of not self is of the form J0(N).

OUTPUT: bool

EXAMPLES:

sage: J0(23).is_J0()
True
sage: J1(11).is_J0()
False
sage: (J0(23) * J1(11)).is_J0()
False
sage: J0(37)[0].is_J0()
False
sage: (J0(23) * J0(21)).is_J0()
False

is_J1()¶

Return whether of not self is of the form J1(N).

OUTPUT: bool

EXAMPLES:

sage: J1(23).is_J1()
True
sage: J0(23).is_J1()
False
sage: (J1(11) * J1(13)).is_J1()
False
sage: (J1(11) * J0(13)).is_J1()
False
sage: J1(23)[0].is_J1()
False

is_ambient()¶

Return True if self equals the ambient product Jacobian.

OUTPUT: bool

EXAMPLES:

sage: A,B,C = J0(33)
sage: A.is_ambient()
False
sage: J0(33).is_ambient()
True
sage: (A+B).is_ambient()
False
sage: (A+B+C).is_ambient()
True

is_hecke_stable()¶

Return True if self is stable under the Hecke operators of its ambient Jacobian.

OUTPUT: bool

EXAMPLES:

sage: J0(11).is_hecke_stable()
True
sage: J0(33)[2].is_hecke_stable()
True
sage: J0(33)[0].is_hecke_stable()
False
sage: (J0(33)[0] + J0(33)[1]).is_hecke_stable()
True

is_simple(none_if_not_known=False)¶

Return whether or not this modular abelian variety is simple, i.e., has no proper nonzero abelian subvarieties.

INPUT:

none_if_not_known - bool (default: False); if True then this function may return None instead of True of False if we don’t already know whether or not self is simple.

EXAMPLES:

sage: J0(5).is_simple(none_if_not_known=True) is None  # this may fail if J0(5) comes up elsewhere...
True
sage: J0(33).is_simple()
False
sage: J0(33).is_simple(none_if_not_known=True)
False
sage: J0(33)[1].is_simple()
True
sage: J1(17).is_simple()
False

is_subvariety(other)¶

Return True if self is a subvariety of other as they sit in a common ambient modular Jacobian. In particular, this function will only return True if self and other have exactly the same ambient Jacobians.

EXAMPLES:

sage: J = J0(37); J
Abelian variety J0(37) of dimension 2
sage: A = J[0]; A
Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37)
sage: A.is_subvariety(A)
True
sage: A.is_subvariety(J)
True

is_subvariety_of_ambient_jacobian()¶

Return True if self is (presented as) a subvariety of the ambient product Jacobian.

Every abelian variety in Sage is a quotient of a subvariety of an ambient Jacobian product by a finite subgroup.

EXAMPLES:

sage: J0(33).is_subvariety_of_ambient_jacobian()
True
sage: A = J0(33)[0]; A
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: A.is_subvariety_of_ambient_jacobian()
True
sage: B, phi = A / A.torsion_subgroup(2)
sage: B
Abelian variety factor of dimension 1 of J0(33)
sage: phi.matrix()
[2 0]
[0 2]
sage: B.is_subvariety_of_ambient_jacobian()
False

isogeny_number(none_if_not_known=False)¶

Return the number (starting at 0) of the isogeny class of new simple abelian varieties that self is in. If self is not simple, raises a ValueError exception.

INPUT:

none_if_not_known - bool (default: False); if True then this function may return None instead of True of False if we don’t already know the isogeny number of self.

EXAMPLES: We test the none_if_not_known flag first:

sage: J0(33).isogeny_number(none_if_not_known=True) is None
True

Of course, \(J_0(33)\) is not simple, so this function raises a ValueError:

sage: J0(33).isogeny_number()
Traceback (most recent call last):
...
ValueError: self must be simple

Each simple factor has isogeny number 1, since that’s the number at which the factor is new.

sage: J0(33)[1].isogeny_number()
0
sage: J0(33)[2].isogeny_number()
0

Next consider \(J_0(37)\) where there are two distinct newform factors:

sage: J0(37)[1].isogeny_number()
1

label()¶

Return the label associated to this modular abelian variety.

The format of the label is [level][isogeny class][group](t, ambient level)

If this abelian variety \(B\) has the above label, this implies only that \(B\) is isogenous to the newform abelian variety \(A_f\) associated to the newform with label [level][isogeny class][group]. The [group] is empty for \(\Gamma_0(N)\), is G1 for \(\Gamma_1(N)\) and is GH[…] for \(\Gamma_H(N)\).

Warning

The sum of \(\delta_s(A_f)\) for all \(s\mid t\) contains \(A\), but no sum for a proper divisor of \(t\) contains \(A\). It need not be the case that \(B\) is equal to \(\delta_t(A_f)\)!!!

OUTPUT: string

EXAMPLES:

sage: J0(11).label()
'11a(1,11)'
sage: J0(11)[0].label()
'11a(1,11)'
sage: J0(33)[2].label()
'33a(1,33)'
sage: J0(22).label()
Traceback (most recent call last):
...
ValueError: self must be simple

We illustrate that self need not equal \(\delta_t(A_f)\):

sage: J = J0(11); phi = J.degeneracy_map(33, 1) + J.degeneracy_map(33,3)
sage: B = phi.image(); B
Abelian subvariety of dimension 1 of J0(33)
sage: B.decomposition()
[
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33)
]
sage: C = J.degeneracy_map(33,3).image(); C
Abelian subvariety of dimension 1 of J0(33)
sage: C == B
False

lattice()¶

Return lattice in ambient cuspidal modular symbols product that defines this modular abelian variety.

This must be defined in each derived class.

OUTPUT: a free module over \(\ZZ\)

EXAMPLES:

sage: A = sage.modular.abvar.abvar.ModularAbelianVariety_abstract((Gamma0(37),), QQ)
sage: A
<repr(<sage.modular.abvar.abvar.ModularAbelianVariety_abstract_with_category at 0x...>) failed: NotImplementedError: BUG -- lattice method must be defined in derived class>

level()¶

Return the level of this modular abelian variety, which is an integer N (usually minimal) such that this modular abelian variety is a quotient of \(J_1(N)\). In the case that the ambient variety of self is a product of Jacobians, return the LCM of their levels.

EXAMPLES:

sage: J1(5077).level()
5077
sage: JH(389,[4]).level()
389
sage: (J0(11)*J0(17)).level()
187

lseries()¶

Return the complex \(L\)-series of this modular abelian variety.

EXAMPLES:

sage: A = J0(37)
sage: A.lseries()
Complex L-series attached to Abelian variety J0(37) of dimension 2

modular_degree()¶

Return the modular degree of this abelian variety, which is the square root of the degree of the modular kernel.

EXAMPLES:

sage: A = AbelianVariety('37a')
sage: A.modular_degree()
2

modular_kernel()¶

Return the modular kernel of this abelian variety, which is the kernel of the canonical polarization of self.

EXAMPLES:

sage: A = AbelianVariety('33a'); A
Newform abelian subvariety 33a of dimension 1 of J0(33)
sage: A.modular_kernel()
Finite subgroup with invariants [3, 3] over QQ of Newform abelian subvariety 33a of dimension 1 of J0(33)

newform(names=None)¶

Return the newform \(f\) such that this abelian variety is isogenous to the newform abelian variety \(A_f\). If this abelian variety is not simple, raise a ValueError.

INPUT:

names – (default: None) If the newform has coefficients in a number field, then a generator name must be specified.

OUTPUT: A newform \(f\) so that self is isogenous to \(A_f\).

EXAMPLES:

sage: J0(11).newform()
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)

sage: f = J0(23).newform(names='a')
sage: AbelianVariety(f) == J0(23)
True

sage: J = J0(33)
sage: [s.newform('a') for s in J.decomposition()]
[q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
 q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
 q + q^2 - q^3 - q^4 - 2*q^5 + O(q^6)]

The following fails since \(J_0(33)\) is not simple:

sage: J0(33).newform()
Traceback (most recent call last):
...
ValueError: self must be simple

newform_decomposition(names=None)¶

Return the newforms of the simple subvarieties in the decomposition of self as a product of simple subvarieties, up to isogeny.

OUTPUT:

an array of newforms

EXAMPLES:

sage: J = J1(11) * J0(23)
sage: J.newform_decomposition('a')
[q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
q + a0*q^2 + (-2*a0 - 1)*q^3 + (-a0 - 1)*q^4 + 2*a0*q^5 + O(q^6)]

newform_label()¶

Return the label [level][isogeny class][group] of the newform \(f\) such that this abelian variety is isogenous to the newform abelian variety \(A_f\). If this abelian variety is not simple, raise a ValueError.

OUTPUT: string

EXAMPLES:

sage: J0(11).newform_label()
'11a'
sage: J0(33)[2].newform_label()
'33a'

The following fails since \(J_0(33)\) is not simple:

sage: J0(33).newform_label()
Traceback (most recent call last):
...
ValueError: self must be simple

newform_level(none_if_not_known=False)¶

Write self as a product (up to isogeny) of newform abelian varieties \(A_f\). Then this function return the least common multiple of the levels of the newforms \(f\), along with the corresponding group or list of groups (the groups do not appear with multiplicity).

INPUT:

none_if_not_known - (default: False) if True, return None instead of attempting to compute the newform level, if it isn’t already known. This None result is not cached.

OUTPUT: integer group or list of distinct groups

EXAMPLES:

sage: J0(33)[0].newform_level()
(11, Congruence Subgroup Gamma0(33))
sage: J0(33)[0].newform_level(none_if_not_known=True)
(11, Congruence Subgroup Gamma0(33))

Here there are multiple groups since there are in fact multiple newforms:

sage: (J0(11) * J1(13)).newform_level()
(143, [Congruence Subgroup Gamma0(11), Congruence Subgroup Gamma1(13)])

number_of_rational_points()¶

Return the number of rational points of this modular abelian variety.

It is not always possible to compute the order of the torsion subgroup. The BSD conjecture is assumed to compute the algebraic rank.

OUTPUT: a positive integer or infinity

EXAMPLES:

sage: J0(23).number_of_rational_points()
11
sage: J0(29).number_of_rational_points()
7
sage: J0(37).number_of_rational_points()
+Infinity

sage: J0(12); J0(12).number_of_rational_points()
Abelian variety J0(12) of dimension 0
1

sage: J1(17).number_of_rational_points()
584

sage: J1(16).number_of_rational_points()
Traceback (most recent call last):
...
RuntimeError: Unable to compute order of torsion subgroup (it is in [1, 2, 4, 5, 10, 20])

padic_lseries(p)¶

Return the \(p\)-adic \(L\)-series of this modular abelian variety.

EXAMPLES:

sage: A = J0(37)
sage: A.padic_lseries(7)
7-adic L-series attached to Abelian variety J0(37) of dimension 2

project_to_factor(n)¶

If self is an ambient product of Jacobians, return a projection from self to the nth such Jacobian.

EXAMPLES:

sage: J = J0(33)
sage: J.project_to_factor(0)
Abelian variety endomorphism of Abelian variety J0(33) of dimension 3

sage: J = J0(33) * J0(37) * J0(11)
sage: J.project_to_factor(2)
Abelian variety morphism:
  From: Abelian variety J0(33) x J0(37) x J0(11) of dimension 6
  To:   Abelian variety J0(11) of dimension 1
sage: J.project_to_factor(2).matrix()
[0 0]
[0 0]
[0 0]
[0 0]
[0 0]
[0 0]
[0 0]
[0 0]
[0 0]
[0 0]
[1 0]
[0 1]

projection(A, check=True)¶

Given an abelian subvariety A of self, return a projection morphism from self to A. Note that this morphism need not be unique.

INPUT:

A - an abelian variety

OUTPUT: a morphism

EXAMPLES:

sage: a,b,c = J0(33)
sage: pi = J0(33).projection(a); pi.matrix()
[ 3 -2]
[-5  5]
[-4  1]
[ 3 -2]
[ 5  0]
[ 1  1]
sage: pi = (a+b).projection(a); pi.matrix()
[ 0  0]
[-3  2]
[-4  1]
[-1 -1]
sage: pi = a.projection(a); pi.matrix()
[1 0]
[0 1]

We project onto a factor in a product of two Jacobians:

sage: A = J0(11)*J0(11); A
Abelian variety J0(11) x J0(11) of dimension 2
sage: A[0]
Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J0(11)
sage: A.projection(A[0])
Abelian variety morphism:
  From: Abelian variety J0(11) x J0(11) of dimension 2
  To:   Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J0(11)
sage: A.projection(A[0]).matrix()
[0 0]
[0 0]
[1 0]
[0 1]
sage: A.projection(A[1]).matrix()
[1 0]
[0 1]
[0 0]
[0 0]

qbar_torsion_subgroup()¶

Return the group of all points of finite order in the algebraic closure of this abelian variety.

EXAMPLES:

sage: T = J0(33).qbar_torsion_subgroup(); T
Group of all torsion points in QQbar on Abelian variety J0(33) of dimension 3

The field of definition is the same as the base field of the abelian variety.

sage: T.field_of_definition()
Rational Field

On the other hand, T is a module over \(\ZZ\).

sage: T.base_ring()
Integer Ring

quotient(other, **kwds)¶

Compute the quotient of self and other, where other is either an abelian subvariety of self or a finite subgroup of self.

INPUT:

other - a finite subgroup or subvariety
further named arguments, that are currently ignored.

OUTPUT: a pair (A, phi) with phi the quotient map from self to A

EXAMPLES: We quotient \(J_0(33)\) out by an abelian subvariety:

sage: Q, f = J0(33).quotient(J0(33)[0])
sage: Q
Abelian variety factor of dimension 2 of J0(33)
sage: f
Abelian variety morphism:
  From: Abelian variety J0(33) of dimension 3
  To:   Abelian variety factor of dimension 2 of J0(33)

We quotient \(J_0(33)\) by the cuspidal subgroup:

sage: C = J0(33).cuspidal_subgroup()
sage: Q, f = J0(33).quotient(C)
sage: Q
Abelian variety factor of dimension 3 of J0(33)
sage: f.kernel()[0]
Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3
sage: C
Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3
sage: J0(11).direct_product(J1(13))
Abelian variety J0(11) x J1(13) of dimension 3

rank()¶

Return the rank of the underlying lattice of self.

EXAMPLES:

sage: J = J0(33)
sage: J.rank()
6
sage: J[1]
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33)
sage: (J[1] * J[1]).rank()
4

rational_cusp_subgroup()¶

Return the subgroup of this modular abelian variety generated by rational cusps.

This is a subgroup of the group of rational points in the cuspidal subgroup.

Warning

This is only currently implemented for \(\Gamma_0(N)\).

EXAMPLES:

sage: J = J0(54)
sage: CQ = J.rational_cusp_subgroup(); CQ
Finite subgroup with invariants [3, 3, 9] over QQ of Abelian variety J0(54) of dimension 4
sage: CQ.gens()
[[(1/3, 0, 0, 1/3, 2/3, 1/3, 0, 1/3)], [(0, 0, 1/9, 1/9, 7/9, 7/9, 1/9, 8/9)], [(0, 0, 0, 0, 0, 0, 1/3, 2/3)]]
sage: factor(CQ.order())
3^4
sage: CQ.invariants()
[3, 3, 9]

In this example the rational cuspidal subgroup and the cuspidal subgroup differ by a lot.

sage: J = J0(49)
sage: J.cuspidal_subgroup()
Finite subgroup with invariants [2, 14] over QQ of Abelian variety J0(49) of dimension 1
sage: J.rational_cusp_subgroup()
Finite subgroup with invariants [2] over QQ of Abelian variety J0(49) of dimension 1

Note that computation of the rational cusp subgroup isn’t implemented for \(\Gamma_1\).

sage: J = J1(13)
sage: J.cuspidal_subgroup()
Finite subgroup with invariants [19, 19] over QQ of Abelian variety J1(13) of dimension 2
sage: J.rational_cusp_subgroup()
Traceback (most recent call last):
...
NotImplementedError: computation of rational cusps only implemented in Gamma0 case.

rational_cuspidal_subgroup()¶

Return the rational subgroup of the cuspidal subgroup of this modular abelian variety.

This is a subgroup of the group of rational points in the cuspidal subgroup.

Warning

This is only currently implemented for \(\Gamma_0(N)\).

EXAMPLES:

sage: J = J0(54)
sage: CQ = J.rational_cuspidal_subgroup(); CQ
Finite subgroup with invariants [3, 3, 9] over QQ of Abelian variety J0(54) of dimension 4
sage: CQ.gens()
[[(1/3, 0, 0, 1/3, 2/3, 1/3, 0, 1/3)], [(0, 0, 1/9, 1/9, 7/9, 7/9, 1/9, 8/9)], [(0, 0, 0, 0, 0, 0, 1/3, 2/3)]]
sage: factor(CQ.order())
3^4
sage: CQ.invariants()
[3, 3, 9]

In this example the rational cuspidal subgroup and the cuspidal subgroup differ by a lot.

sage: J = J0(49)
sage: J.cuspidal_subgroup()
Finite subgroup with invariants [2, 14] over QQ of Abelian variety J0(49) of dimension 1
sage: J.rational_cuspidal_subgroup()
Finite subgroup with invariants [2] over QQ of Abelian variety J0(49) of dimension 1

Note that computation of the rational cusp subgroup isn’t implemented for \(\Gamma_1\).

sage: J = J1(13)
sage: J.cuspidal_subgroup()
Finite subgroup with invariants [19, 19] over QQ of Abelian variety J1(13) of dimension 2
sage: J.rational_cuspidal_subgroup()
Traceback (most recent call last):
...
NotImplementedError: only implemented when group is Gamma0

rational_homology()¶

Return the rational homology of this modular abelian variety.

EXAMPLES:

sage: H = J0(37).rational_homology(); H
Rational Homology of Abelian variety J0(37) of dimension 2
sage: H.rank()
4
sage: H.base_ring()
Rational Field
sage: H = J1(17).rational_homology(); H
Rational Homology of Abelian variety J1(17) of dimension 5
sage: H.rank()
10
sage: H.base_ring()
Rational Field

rational_torsion_order(proof=True)¶

Return the order of the rational torsion subgroup of this modular abelian variety.

This function is really an alias for order() See the docstring there for a more in-depth reference and more interesting examples.

INPUT:

proof – a boolean (default: True)

OUTPUT:

The order of the rational torsion subgroup of this modular abelian variety.

EXAMPLES:

sage: J0(11).rational_torsion_subgroup().order()
5
sage: J0(11).rational_torsion_order()
5

rational_torsion_subgroup()¶

Return the maximal torsion subgroup of self defined over QQ.

EXAMPLES:

sage: J = J0(33)
sage: A = J.new_subvariety()
sage: A
Abelian subvariety of dimension 1 of J0(33)
sage: t = A.rational_torsion_subgroup(); t
Torsion subgroup of Abelian subvariety of dimension 1 of J0(33)
sage: t.multiple_of_order()
4
sage: t.divisor_of_order()
4
sage: t.order()
4
sage: t.gens()
[[(1/2, 0, 0, -1/2, 0, 0)], [(0, 0, 1/2, 0, 1/2, -1/2)]]

shimura_subgroup()¶

Return the Shimura subgroup of this modular abelian variety. This is the kernel of \(J_0(N) \rightarrow J_1(N)\) under the natural map. Here we compute the Shimura subgroup as the kernel of \(J_0(N) \rightarrow J_0(Np)\) where the map is the difference between the two degeneracy maps.

EXAMPLES:

sage: J=J0(11)
sage: J.shimura_subgroup()
Finite subgroup with invariants [5] over QQ of Abelian variety J0(11) of dimension 1

sage: J=J0(17)
sage: G=J.cuspidal_subgroup(); G
Finite subgroup with invariants [4] over QQ of Abelian variety J0(17) of dimension 1
sage: S=J.shimura_subgroup(); S
Finite subgroup with invariants [4] over QQ of Abelian variety J0(17) of dimension 1
sage: G.intersection(S)
Finite subgroup with invariants [2] over QQ of Abelian variety J0(17) of dimension 1

sage: J=J0(33)
sage: A=J.decomposition()[0]
sage: A.shimura_subgroup()
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: J.shimura_subgroup()
Finite subgroup with invariants [10] over QQ of Abelian variety J0(33) of dimension 3

sturm_bound()¶

Return a bound \(B\) such that all Hecke operators \(T_n\) for \(n\leq B\) generate the Hecke algebra.

OUTPUT: integer

EXAMPLES:

sage: J0(11).sturm_bound()
2
sage: J0(33).sturm_bound()
8
sage: J1(17).sturm_bound()
48
sage: J1(123456).sturm_bound()
1693483008
sage: JH(37,[2,3]).sturm_bound()
7
sage: J1(37).sturm_bound()
228

torsion_subgroup(n)¶

If n is an integer, return the subgroup of points of order n. Return the \(n\)-torsion subgroup of elements of order dividing \(n\) of this modular abelian variety \(A\), i.e., the group \(A[n]\).

EXAMPLES:

sage: J1(13).torsion_subgroup(19)
Finite subgroup with invariants [19, 19, 19, 19] over QQ of Abelian variety J1(13) of dimension 2

sage: A = J0(23)
sage: G = A.torsion_subgroup(5); G
Finite subgroup with invariants [5, 5, 5, 5] over QQ of Abelian variety J0(23) of dimension 2
sage: G.order()
625
sage: G.gens()
[[(1/5, 0, 0, 0)], [(0, 1/5, 0, 0)], [(0, 0, 1/5, 0)], [(0, 0, 0, 1/5)]]
sage: A = J0(23)
sage: A.torsion_subgroup(2).order()
16

vector_space()¶

Return vector space corresponding to the modular abelian variety.

This is the lattice tensored with \(\QQ\).

EXAMPLES:

sage: J0(37).vector_space()
Vector space of dimension 4 over Rational Field
sage: J0(37)[0].vector_space()
Vector space of degree 4 and dimension 2 over Rational Field
Basis matrix:
[   1   -1    0  1/2]
[   0    0    1 -1/2]

zero_subgroup()¶

Return the zero subgroup of this modular abelian variety, as a finite group.

EXAMPLES:

sage: A =J0(54); G = A.zero_subgroup(); G
Finite subgroup with invariants [] over QQ of Abelian variety J0(54) of dimension 4
sage: G.is_subgroup(A)
True

zero_subvariety()¶

Return the zero subvariety of self.

EXAMPLES:

sage: J = J0(37)
sage: J.zero_subvariety()
Simple abelian subvariety of dimension 0 of J0(37)
sage: J.zero_subvariety().level()
37
sage: J.zero_subvariety().newform_level()
(1, [])

class sage.modular.abvar.abvar.ModularAbelianVariety_modsym(modsym, lattice=None, newform_level=None, is_simple=None, isogeny_number=None, number=None, check=True)¶

Bases: sage.modular.abvar.abvar.ModularAbelianVariety_modsym_abstract

Modular abelian variety that corresponds to a Hecke stable space of cuspidal modular symbols.

EXAMPLES: We create a modular abelian variety attached to a space of modular symbols.

sage: M = ModularSymbols(23).cuspidal_submodule()
sage: A = M.abelian_variety(); A
Abelian variety J0(23) of dimension 2

brandt_module(p)¶

Return the Brandt module at p that corresponds to self. This is the factor of the vector space on the ideal class set in an order of level N in the quaternion algebra ramified at p and infinity.

INPUT:

p – prime that exactly divides the level

OUTPUT:

Brandt module space that corresponds to self.

EXAMPLES:

sage: J0(43)[1].brandt_module(43)
Subspace of dimension 2 of Brandt module of dimension 4 of level 43 of weight 2 over Rational Field
sage: J0(43)[1].brandt_module(43).basis()
((1, 0, -1/2, -1/2), (0, 1, -1/2, -1/2))
sage: J0(43)[0].brandt_module(43).basis()
((0, 0, 1, -1),)
sage: J0(35)[0].brandt_module(5).basis()
((1, 0, -1, 0),)
sage: J0(35)[0].brandt_module(7).basis()
((1, -1, 1, -1),)

component_group_order(p)¶

Return the order of the component group of the special fiber at p of the Neron model of self.

NOTE: For bad primes, this is only implemented when the group if Gamma0 and p exactly divides the level.

NOTE: the input abelian variety must be simple

ALGORITHM: See “Component Groups of Quotients of J0(N)” by Kohel and Stein. That paper is about optimal quotients; however, section 4.1 of Conrad-Stein “Component Groups of Purely Toric Quotients”, one sees that the component group of an optimal quotient is the same as the component group of its dual (which is the subvariety).

INPUT:

p – a prime number

OUTPUT:

Integer

EXAMPLES:

sage: A = J0(37)[1]
sage: A.component_group_order(37)
3
sage: A = J0(43)[1]
sage: A.component_group_order(37)
1
sage: A.component_group_order(43)
7
sage: A = J0(23)[0]
sage: A.component_group_order(23)
11

tamagawa_number(p)¶

Return the Tamagawa number of this abelian variety at p.

NOTE: For bad primes, this is only implemented when the group if Gamma0 and p exactly divides the level and Atkin-Lehner acts diagonally on this abelian variety (e.g., if this variety is new and simple). See the self.component_group command for more information.

NOTE: the input abelian variety must be simple

In cases where this function doesn’t work, consider using the self.tamagawa_number_bounds functions.

INPUT:

p – a prime number

OUTPUT:

Integer

EXAMPLES:

sage: A = J0(37)[1]
sage: A.tamagawa_number(37)
3
sage: A = J0(43)[1]
sage: A.tamagawa_number(37)
1
sage: A.tamagawa_number(43)
7
sage: A = J0(23)[0]
sage: A.tamagawa_number(23)
11

tamagawa_number_bounds(p)¶

Return a divisor and multiple of the Tamagawa number of self at \(p\).

NOTE: the input abelian variety must be simple.

INPUT:

\(p\) – a prime number

OUTPUT:

div – integer; divisor of Tamagawa number at \(p\)
mul – integer; multiple of Tamagawa number at \(p\)
mul_primes – tuple; in case mul==0, a list of all primes that can possibly divide the Tamagawa number at \(p\)

EXAMPLES:

sage: A = J0(63).new_subvariety()[1]; A
Simple abelian subvariety 63b(1,63) of dimension 2 of J0(63)
sage: A.tamagawa_number_bounds(7)
(3, 3, ())
sage: A.tamagawa_number_bounds(3)
(1, 0, (2, 3, 5))

class sage.modular.abvar.abvar.ModularAbelianVariety_modsym_abstract(groups, base_field, is_simple=None, newform_level=None, isogeny_number=None, number=None, check=True)¶

Bases: sage.modular.abvar.abvar.ModularAbelianVariety_abstract

decomposition(simple=True, bound=None)¶

Decompose this modular abelian variety as a product of abelian subvarieties, up to isogeny.

INPUT: simple- bool (default: True) if True, all factors are simple. If False, each factor returned is isogenous to a power of a simple and the simples in each factor are distinct.

bound - int (default: None) if given, only use Hecke operators up to this bound when decomposing. This can give wrong answers, so use with caution!

EXAMPLES:

sage: J = J0(33)
sage: J.decomposition()
[
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33),
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33),
Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
]
sage: J1(17).decomposition()
[
Simple abelian subvariety 17aG1(1,17) of dimension 1 of J1(17),
Simple abelian subvariety 17bG1(1,17) of dimension 4 of J1(17)
]

dimension()¶

Return the dimension of this modular abelian variety.

EXAMPLES:

sage: J0(37)[0].dimension()
1
sage: J0(43)[1].dimension()
2
sage: J1(17)[1].dimension()
4

group()¶

Return the congruence subgroup associated that this modular abelian variety is associated to.

EXAMPLES:

sage: J0(13).group()
Congruence Subgroup Gamma0(13)
sage: J1(997).group()
Congruence Subgroup Gamma1(997)
sage: JH(37,[3]).group()
Congruence Subgroup Gamma_H(37) with H generated by [3]
sage: J0(37)[1].groups()
(Congruence Subgroup Gamma0(37),)

groups()¶

Return the tuple of groups associated to the modular symbols abelian variety. This is always a 1-tuple.

OUTPUT: tuple

EXAMPLES:

sage: A = ModularSymbols(33).cuspidal_submodule().abelian_variety(); A
Abelian variety J0(33) of dimension 3
sage: A.groups()
(Congruence Subgroup Gamma0(33),)
sage: type(A)
<class 'sage.modular.abvar.abvar.ModularAbelianVariety_modsym_with_category'>

is_ambient()¶

Return True if this abelian variety attached to a modular symbols space is attached to the cuspidal subspace of the ambient modular symbols space.

OUTPUT: bool

EXAMPLES:

sage: A = ModularSymbols(43).cuspidal_subspace().abelian_variety(); A
Abelian variety J0(43) of dimension 3
sage: A.is_ambient()
True
sage: type(A)
<class 'sage.modular.abvar.abvar.ModularAbelianVariety_modsym_with_category'>
sage: A = ModularSymbols(43).cuspidal_subspace()[1].abelian_variety(); A
Abelian subvariety of dimension 2 of J0(43)
sage: A.is_ambient()
False

is_subvariety(other)¶

Return True if self is a subvariety of other.

EXAMPLES:

sage: J = J0(37); J
Abelian variety J0(37) of dimension 2
sage: A = J[0]; A
Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37)
sage: A.is_subvariety(J)
True
sage: A.is_subvariety(J0(11))
False

There may be a way to map \(A\) into \(J_0(74)\), but \(A\) is not equipped with any special structure of an embedding.

sage: A.is_subvariety(J0(74))
False

Some ambient examples:

sage: J = J0(37)
sage: J.is_subvariety(J)
True
sage: J.is_subvariety(25)
False

More examples:

sage: A = J0(42); D = A.decomposition(); D
[
Simple abelian subvariety 14a(1,42) of dimension 1 of J0(42),
Simple abelian subvariety 14a(3,42) of dimension 1 of J0(42),
Simple abelian subvariety 21a(1,42) of dimension 1 of J0(42),
Simple abelian subvariety 21a(2,42) of dimension 1 of J0(42),
Simple abelian subvariety 42a(1,42) of dimension 1 of J0(42)
]
sage: D[0].is_subvariety(A)
True
sage: D[1].is_subvariety(D[0] + D[1])
True
sage: D[2].is_subvariety(D[0] + D[1])
False

lattice()¶

Return the lattice defining this modular abelian variety.

OUTPUT:

A free \(\ZZ\)-module embedded in an ambient \(\QQ\)-vector space.

EXAMPLES:

sage: A = ModularSymbols(33).cuspidal_submodule()[0].abelian_variety(); A
Abelian subvariety of dimension 1 of J0(33)
sage: A.lattice()
Free module of degree 6 and rank 2 over Integer Ring
User basis matrix:
[ 1  0  0 -1  0  0]
[ 0  0  1  0  1 -1]
sage: type(A)
<class 'sage.modular.abvar.abvar.ModularAbelianVariety_modsym_with_category'>

modular_symbols(sign=0)¶

Return space of modular symbols (with given sign) associated to this modular abelian variety, if it can be found by cutting down using Hecke operators. Otherwise raise a RuntimeError exception.

EXAMPLES:

sage: A = J0(37)
sage: A.modular_symbols()
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field
sage: A.modular_symbols(1)
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(37) of weight 2 with sign 1 over Rational Field

More examples:

sage: J0(11).modular_symbols()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: J0(11).modular_symbols(sign=1)
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
sage: J0(11).modular_symbols(sign=0)
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: J0(11).modular_symbols(sign=-1)
Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field

Even more examples:

sage: A = J0(33)[1]; A
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33)
sage: A.modular_symbols()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field

It is not always possible to determine the sign subspaces:

sage: A.modular_symbols(1)
Traceback (most recent call last):
...
RuntimeError: unable to determine sign (=1) space of modular symbols

sage: A.modular_symbols(-1)
Traceback (most recent call last):
...
RuntimeError: unable to determine sign (=-1) space of modular symbols

new_subvariety(p=None)¶

Return the new or \(p\)-new subvariety of self.

INPUT:

self - a modular abelian variety
p - prime number or None (default); if p is a prime, return the p-new subvariety. Otherwise return the full new subvariety.

EXAMPLES:

sage: J0(34).new_subvariety()
Abelian subvariety of dimension 1 of J0(34)
sage: J0(100).new_subvariety()
Abelian subvariety of dimension 1 of J0(100)
sage: J1(13).new_subvariety()
Abelian variety J1(13) of dimension 2

old_subvariety(p=None)¶

Return the old or \(p\)-old abelian variety of self.

INPUT:

self - a modular abelian variety
p - prime number or None (default); if p is a prime, return the p-old subvariety. Otherwise return the full old subvariety.

EXAMPLES:

sage: J0(33).old_subvariety()
Abelian subvariety of dimension 2 of J0(33)
sage: J0(100).old_subvariety()
Abelian subvariety of dimension 6 of J0(100)
sage: J1(13).old_subvariety()
Abelian subvariety of dimension 0 of J1(13)

sage.modular.abvar.abvar.factor_modsym_space_new_factors(M)¶

Return the factorizations of all the new subspaces of \(M\).

INPUT:

M – ambient modular symbols space

OUTPUT: list of decompositions corresponding to each new space.

EXAMPLES:

sage: M = ModularSymbols(33)
sage: sage.modular.abvar.abvar.factor_modsym_space_new_factors(M)
[[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
],
 [
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field
]]

sage.modular.abvar.abvar.factor_new_space(M)¶

Given a new space \(M\) of modular symbols, return the decomposition into simple of \(M\) under the Hecke operators.

INPUT:

M - modular symbols space

OUTPUT: list of factors

EXAMPLES:

sage: M = ModularSymbols(37).cuspidal_subspace()
sage: sage.modular.abvar.abvar.factor_new_space(M)
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field
]

sage.modular.abvar.abvar.is_ModularAbelianVariety(x)¶

Return True if x is a modular abelian variety.

INPUT:

x - object

EXAMPLES:

sage: from sage.modular.abvar.abvar import is_ModularAbelianVariety
sage: is_ModularAbelianVariety(5)
False
sage: is_ModularAbelianVariety(J0(37))
True

Returning True is a statement about the data type not whether or not some abelian variety is modular:

sage: is_ModularAbelianVariety(EllipticCurve('37a'))
False

sage.modular.abvar.abvar.modsym_lattices(M, factors)¶

Append lattice information to the output of simple_factorization_of_modsym_space.

INPUT:

M - modular symbols spaces
factors - Sequence (simple_factorization_of_modsym_space)

OUTPUT: sequence with more information for each factor (the lattice)

EXAMPLES:

sage: M = ModularSymbols(33)
sage: factors = sage.modular.abvar.abvar.simple_factorization_of_modsym_space(M, simple=False)
sage: sage.modular.abvar.abvar.modsym_lattices(M, factors)
[
(11, 0, None, Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field, Free module of degree 6 and rank 4 over Integer Ring
Echelon basis matrix:
[ 1  0  0  0 -1  2]
[ 0  1  0  0 -1  1]
[ 0  0  1  0 -2  2]
[ 0  0  0  1 -1 -1]),
(33, 0, None, Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field, Free module of degree 6 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1  0  0 -1  0  0]
[ 0  0  1  0  1 -1])
]

sage.modular.abvar.abvar.random_hecke_operator(M, t=None, p=2)¶

Return a random Hecke operator acting on \(M\), got by adding to \(t\) a random multiple of \(T_p\)

INPUT:

M - modular symbols space
t - None or a Hecke operator
p - a prime

OUTPUT: Hecke operator prime

EXAMPLES:

sage: M = ModularSymbols(11).cuspidal_subspace()
sage: t, p = sage.modular.abvar.abvar.random_hecke_operator(M)
sage: p
3
sage: t, p = sage.modular.abvar.abvar.random_hecke_operator(M, t, p)
sage: p
5

sage.modular.abvar.abvar.simple_factorization_of_modsym_space(M, simple=True)¶

Return the canonical factorization of \(M\) into (simple) subspaces.

INPUT:

M – ambient modular symbols space
simple – boolean (default: True); if set to False, isogenous simple factors are grouped together

OUTPUT: sequence

EXAMPLES:

sage: M = ModularSymbols(33)
sage: sage.modular.abvar.abvar.simple_factorization_of_modsym_space(M)
[
(11, 0, 1, Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field),
(11, 0, 3, Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field),
(33, 0, 1, Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field)
]
sage: sage.modular.abvar.abvar.simple_factorization_of_modsym_space(M, simple=False)
[
(11, 0, None, Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field),
(33, 0, None, Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field)
]

sage.modular.abvar.abvar.sqrt_poly(f)¶

Return the square root of the polynomial \(f\).

Note

At some point something like this should be a member of the polynomial class. For now this is just used internally by some charpoly functions above.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = (x-1)*(x+2)*(x^2 + 1/3*x + 5)
sage: f
x^4 + 4/3*x^3 + 10/3*x^2 + 13/3*x - 10
sage: sage.modular.abvar.abvar.sqrt_poly(f^2)
x^4 + 4/3*x^3 + 10/3*x^2 + 13/3*x - 10
sage: sage.modular.abvar.abvar.sqrt_poly(f)
Traceback (most recent call last):
...
ValueError: f must be a perfect square
sage: sage.modular.abvar.abvar.sqrt_poly(2*f^2)
Traceback (most recent call last):
...
ValueError: f must be monic

Base class for modular abelian varieties¶

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