43.2 Base class for modular abelian varieties

Module: sage.modular.abvar.abvar

Base class for modular abelian varieties

Author: William Stein (2007-03)

TESTS:

sage: A = J0(33)
sage: D = A.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: loads(dumps(D)) == D
True
sage: loads(dumps(A)) == A
True

Module-level Functions

factor_modsym_space_new_factors( M)

Given an ambient modular symbols space, return complete factorization of it.

Input:

M
- modular symbols space
Output: list of decompositions corresponding to each new space.

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
]]

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

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
]

is_ModularAbelianVariety( x)

Return True if x is a modular abelian variety.

Input:

x
- object

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

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)

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])
]

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

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

simple_factorization_of_modsym_space( M, [simple=True])

Return factorization of $ M$ . If simple is False, return powers of simples.

Input:

M
- modular symbols space
simple
- bool (default: True)
Output: sequence

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)
]

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.

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

Class: ModularAbelianVariety

class ModularAbelianVariety
ModularAbelianVariety( self, groups, [lattice=None], [base_field=Rational Field], [is_simple=None], [newform_level=None], [isogeny_number=None], [number=None], [check=True])

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

Input:

groups
- a tuple of congruence subgroups
lattice
- (default: $ \mathbf{Z}^n$ ) a full lattice in $ \mathbf{Z}^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: $ \mathbf{Q}$ )

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

Functions: lattice

lattice( self)

Return the lattice that defines this abelan variety.

Output:

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

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'>
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]

Special Functions: __init__

Class: ModularAbelianVariety_abstract

class ModularAbelianVariety_abstract
ModularAbelianVariety_abstract( self, groups, base_field, [is_simple=None], [newform_level=None], [isogeny_number=None], [number=None], [check=True])

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 isogeneous 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

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'>

All hell breaks loose if you try to do anything with $ A$ :

sage: A
Traceback (most recent call last):
...
NotImplementedError: BUG -- lattice method must be defined in derived class

Functions: ambient_morphism,$ \,$ ambient_variety,$ \,$ base_extend,$ \,$ base_field,$ \,$ category,$ \,$ change_ring,$ \,$ complement,$ \,$ cuspidal_subgroup,$ \,$ decomposition,$ \,$ degen_t,$ \,$ degeneracy_map,$ \,$ degree,$ \,$ dimension,$ \,$ direct_product,$ \,$ dual,$ \,$ endomorphism_ring,$ \,$ finite_subgroup,$ \,$ free_module,$ \,$ groups,$ \,$ hecke_operator,$ \,$ hecke_polynomial,$ \,$ homology,$ \,$ in_same_ambient_variety,$ \,$ integral_homology,$ \,$ intersection,$ \,$ is_ambient,$ \,$ is_hecke_stable,$ \,$ is_simple,$ \,$ is_subvariety,$ \,$ is_subvariety_of_ambient_jacobian,$ \,$ isogeny_number,$ \,$ label,$ \,$ lattice,$ \,$ level,$ \,$ lseries,$ \,$ modular_degree,$ \,$ modular_kernel,$ \,$ newform_label,$ \,$ newform_level,$ \,$ padic_lseries,$ \,$ project_to_factor,$ \,$ projection,$ \,$ qbar_torsion_subgroup,$ \,$ quotient,$ \,$ rank,$ \,$ rational_cusp_subgroup,$ \,$ rational_homology,$ \,$ rational_torsion_subgroup,$ \,$ sturm_bound,$ \,$ torsion_subgroup,$ \,$ vector_space,$ \,$ zero_subgroup,$ \,$ zero_subvariety

ambient_morphism( self)

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.

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-injecture 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( self)

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

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( self, K)

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( self)

Synonym for self.base_ring().

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

category( self)

Return the category of modular abelian varieties that contains this modular abelian variety.

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

change_ring( self, R)

Change the base ring of this modular abelian variety.

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

complement( self, [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

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)

cuspidal_subgroup( self)

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

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( self, [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!

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( self, [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)

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( self, 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

We make several degenerancy 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 complciated 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( self)

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

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

dimension( self)

Return the dimension of this abelian variety.

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

direct_product( self, other)

Compute the direct product of self and other.

Input:

self, other
- modular abelian varieties

Output: abelian variety

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( self)

Return the dual of this abelian variety.

Output: abelian variety

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.

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 $ (\mathbf{Z}/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 = 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 = 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.

endomorphism_ring( self)

Return the endomorphism ring of self.

Output: b = self.sturm_bound()

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( self, 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

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

free_module( self)

Synonym for self.lattice().

Output: a free module over $ \mathbf{Z}$

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]

groups( self)

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$ .

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( self, n)

Return the $ n$ -th Hecke operator on the modular abelian variety, if this makes sense [[elaborate]]. Otherwise raise a ValueError.

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( self, 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

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( self, [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.

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( self, other)

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

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( self)

Return the integral homology of this modular abelian variety.

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 -2  0 -1  1]
[ 0  0  1 -1  0 -1]
[ 0  0  1  1 -2  0]
[ 0  0  0  1 -1 -1]

intersection( self, other)

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

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

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))

is_ambient( self)

Return True if self equals the ambient product Jacobian.

Output: bool

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( self)

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

Output: bool

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( self, [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.

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( self, 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.

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( self)

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.

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( self, [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.

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( self)

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 abelan 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

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( self)

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 $ \mathbf{Z}$

sage: A = sage.modular.abvar.abvar.ModularAbelianVariety_abstract((Gamma0(37),), QQ)
sage: A
Traceback (most recent call last):
...
NotImplementedError: BUG -- lattice method must be defined in derived class

level( self)

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.

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

lseries( self)

Return the complex $ L$ -series of this modular abelian variety.

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

modular_degree( self)

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

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

modular_kernel( self)

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

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_label( self)

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

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( self, [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

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)])

padic_lseries( self, p)

Return the $ p$ -adic $ L$ -series of this modular abelian variety.

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( self, n)

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

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( self, 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

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( self)

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

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 $ \mathbf{Z}$ .

sage: T.base_ring()
Integer Ring

quotient( self, other)

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

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

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( self)

Return the rank of the underlying lattice of self.

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( self)

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)$ .

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_homology( self)

Return the rational homology of this modular abelian variety.

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_subgroup( self)

Return the maximal torsion subgroup of self defined over QQ.

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()
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)]]
sage: t
Torsion subgroup of Abelian subvariety of dimension 1 of J0(33)

sturm_bound( self)

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

Output: integer

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()
252

torsion_subgroup( self, 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]$ .

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( self)

Return vector space corresponding to the modular abelian variety.

This is the lattice tensored with $ \mathbf{Q}$ .

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( self)

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

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( self)

Return the zero subvariety of self.

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, [])

Special Functions: __add__,$ \,$ __cmp__,$ \,$ __contains__,$ \,$ __div__,$ \,$ __getitem__,$ \,$ __getslice__,$ \,$ __init__,$ \,$ __mul__,$ \,$ __pow__,$ \,$ __radd__,$ \,$ _ambient_cuspidal_subgroup,$ \,$ _ambient_dimension,$ \,$ _ambient_hecke_matrix_on_modular_symbols,$ \,$ _ambient_latex_repr,$ \,$ _ambient_lattice,$ \,$ _ambient_modular_symbols_abvars,$ \,$ _ambient_modular_symbols_spaces,$ \,$ _ambient_repr,$ \,$ _classify_ambient_factors,$ \,$ _complement_shares_no_factors_with_same_label,$ \,$ _compute_hecke_polynomial,$ \,$ _factors_with_same_label,$ \,$ _Hom_,$ \,$ _integral_hecke_matrix,$ \,$ _isogeny_to_newform_abelian_variety,$ \,$ _isogeny_to_product_of_powers,$ \,$ _isogeny_to_product_of_simples,$ \,$ _quotient_by_abelian_subvariety,$ \,$ _quotient_by_finite_subgroup,$ \,$ _rational_hecke_matrix,$ \,$ _rational_homology_space,$ \,$ _repr_,$ \,$ _simple_isogeny

__add__( self, other)

Returns the sum of the images of self and other inside the ambient Jacobian product. self and other must be abelian subvarieties of the ambient Jacobian product.

WARNING: The sum of course only makes sense in some ambient variety, and by definition this function takes the sum of the images of both self and other in the ambient product Jacobian.

We compute the sum of two abelian varieties of $ J_0(33)$ :

sage: J = J0(33)
sage: J[0] + J[1]
Abelian subvariety of dimension 2 of J0(33)

We sum all three and get the full $ J_0(33)$ :

sage: (J[0] + J[1]) + (J[1] + J[2])
Abelian variety J0(33) of dimension 3

Adding to zero works:

sage: J[0] + 0
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)

Hence the sum command works:

sage: sum([J[0], J[2]])
Abelian subvariety of dimension 2 of J0(33)

We try to add something in $ J_0(33)$ to something in $ J_0(11)$ ; this shouldn't and doesn't work.

sage: J[0] + J0(11)
Traceback (most recent call last):
...
TypeError: sum not defined since ambient spaces different

We compute the diagonal image of $ J_0(11)$ in $ J_0(33)$ , then add the result to the new elliptic curve of level $ 33$ .

sage: A = J0(11)
sage: B = (A.degeneracy_map(33,1) + A.degeneracy_map(33,3)).image()
sage: B + J0(33)[2]
Abelian subvariety of dimension 2 of J0(33)

__cmp__( self, other)

Compare two modular abelian varieties.

If other is not a modular abelian variety, compares the types of self and other. If other is a modular abelian variety, compares the groups, then if those are the same, compares the newform level and isogeny class number and degeneracy map numbers. If those are not defined or matched up, compare the underlying lattices.

sage: cmp(J0(37)[0], J0(37)[1])
-1
sage: cmp(J0(33)[0], J0(33)[1])
-1
sage: cmp(J0(37), 5) #random
1

__contains__( self, x)

Determine whether or not self contains x.

sage: J = J0(67); G = (J[0] + J[1]).intersection(J[1] + J[2])
sage: G[0]
Finite subgroup with invariants [5, 10] over QQbar of Abelian subvariety of
dimension 3 of J0(67)
sage: a = G[0].0; a
[(1/10, 1/10, 3/10, 1/2, 1, -2, -3, 33/10, 0, -1/2)]
sage: a in J[0]
False
sage: a in (J[0]+J[1])
True
sage: a in (J[1]+J[2])
True
sage: C = G[1]   # abelian variety in kernel
sage: G[0].0
[(1/10, 1/10, 3/10, 1/2, 1, -2, -3, 33/10, 0, -1/2)]
sage: 5*G[0].0
[(1/2, 1/2, 3/2, 5/2, 5, -10, -15, 33/2, 0, -5/2)]
sage: 5*G[0].0 in C
True

__div__( self, other)

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

Quotient out by a finite group:

sage: J = J0(67); G = (J[0] + J[1]).intersection(J[1] + J[2])
sage: Q, _ = J/G[0]; Q
Abelian variety factor of dimension 5 of J0(67) over Algebraic Field
sage: Q.base_field()
Algebraic Field
sage: Q.lattice()
Free module of degree 10 and rank 10 over Integer Ring
Echelon basis matrix:
[1/10 1/10 3/10  1/2    0    0    0 3/10    0  1/2]
[   0  1/5  4/5  4/5    0    0    0    0    0  3/5]
...

Quotient out by an abelian subvariety:

sage: A, B, C = J0(33)
sage: Q, phi = J0(33)/A
sage: Q
Abelian variety factor of dimension 2 of J0(33)
sage: phi.domain()
Abelian variety J0(33) of dimension 3
sage: phi.codomain()
Abelian variety factor of dimension 2 of J0(33)
sage: phi.kernel()
(Finite subgroup with invariants [2] over QQbar of Abelian variety J0(33)
of dimension 3,
 Abelian subvariety of dimension 1 of J0(33))
sage: phi.kernel()[1] == A
True

The abelian variety we quotient out by must be an abelian subvariety.

sage: Q = (A + B)/C; Q
Traceback (most recent call last):
...
TypeError: other must be a subgroup or abelian subvariety

__getitem__( self, i)

Return the i-th decomposition factor of self.

sage: J = J0(389)
sage: J.decomposition()
[
Simple abelian subvariety 389a(1,389) of dimension 1 of J0(389),
Simple abelian subvariety 389b(1,389) of dimension 2 of J0(389),
Simple abelian subvariety 389c(1,389) of dimension 3 of J0(389),
Simple abelian subvariety 389d(1,389) of dimension 6 of J0(389),
Simple abelian subvariety 389e(1,389) of dimension 20 of J0(389)
]
sage: J[2]
Simple abelian subvariety 389c(1,389) of dimension 3 of J0(389)
sage: J[-1]
Simple abelian subvariety 389e(1,389) of dimension 20 of J0(389)

__getslice__( self, i, j)

The slice i:j of decompositions of self.

sage: J = J0(125); J.decomposition()
[
Simple abelian subvariety 125a(1,125) of dimension 2 of J0(125),
Simple abelian subvariety 125b(1,125) of dimension 2 of J0(125),
Simple abelian subvariety 125c(1,125) of dimension 4 of J0(125)
]
sage: J[:2]
[
Simple abelian subvariety 125a(1,125) of dimension 2 of J0(125),
Simple abelian subvariety 125b(1,125) of dimension 2 of J0(125)
]

__mul__( self, other)

Compute the direct product of self and other.

Some modular Jacobians:

sage: J0(11) * J0(33)
Abelian variety J0(11) x J0(33) of dimension 4
sage: J0(11) * J0(33) * J0(11)
Abelian variety J0(11) x J0(33) x J0(11) of dimension 5

We multiply some factors of $ J_0(65)$ :

sage: d = J0(65).decomposition()
sage: d[0] * d[1] * J0(11)
Abelian subvariety of dimension 4 of J0(65) x J0(65) x J0(11)

__pow__( self, n)

Return $ n$ th power of self.

Input:

n
- a nonnegative integer
Output: an abelian variety

sage: J = J0(37)
sage: J^0
Simple abelian subvariety of dimension 0 of J0(37)
sage: J^1
Abelian variety J0(37) of dimension 2
sage: J^1 is J
True

__radd__( self, other)

Return other + self when other is 0. Otherwise raise a TypeError.

sage: int(0) + J0(37)
Abelian variety J0(37) of dimension 2

_ambient_cuspidal_subgroup( self, [rational_only=False])

sage: (J1(13)*J0(11))._ambient_cuspidal_subgroup()
Finite subgroup with invariants [19, 95] over QQ of Abelian variety J1(13)
x J0(11) of dimension 3
sage: (J0(33))._ambient_cuspidal_subgroup()
Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33)
of dimension 3
sage: (J0(33)*J0(33))._ambient_cuspidal_subgroup()
Finite subgroup with invariants [10, 10, 10, 10] over QQ of Abelian variety
J0(33) x J0(33) of dimension 6

_ambient_dimension( self)

Return the dimension of the ambient Jacobian product.

sage: A = J0(37) * J1(13); A
Abelian variety J0(37) x J1(13) of dimension 4
sage: A._ambient_dimension()
4
sage: B = A[0]; B
Simple abelian subvariety 13aG1(1,13) of dimension 2 of J0(37) x J1(13)
sage: B._ambient_dimension()
4

This example is fast because it implicitly calls _ambient_dimension.

sage: J0(902834082394)
Abelian variety J0(902834082394) of dimension 113064825881

_ambient_hecke_matrix_on_modular_symbols( self, n)

Return block direct sum of the matrix of the Hecke operator $ T_n$ acting on each of the ambient modular symbols spaces.

Input:

n
- an integer $ \geq 1$ .

Output: a matrix

sage: (J0(11) * J1(13))._ambient_hecke_matrix_on_modular_symbols(2)
[-2  0  0  0  0  0]
[ 0 -2  0  0  0  0]
[ 0  0 -2  0 -1  1]
[ 0  0  1 -1  0 -1]
[ 0  0  1  1 -2  0]
[ 0  0  0  1 -1 -1]

_ambient_latex_repr( self)

Return Latex representation of the ambient product.

Output: string

sage: (J0(11) * J0(33))._ambient_latex_repr()
'J_0(11) \cross J_0(33)'

_ambient_lattice( self)

Return free lattice of rank twice the degree of self. This is the lattice corresponding to the ambient product Jacobian.

Output: lattice

We compute the ambient lattice of a product:

sage: (J0(33)*J1(11))._ambient_lattice()
Ambient free module of rank 8 over the principal ideal domain Integer Ring

We compute the ambient lattice of an abelian subvariety $ J_0(33)$ , which is the same as the lattice for the $ J_0(33)$ itself:

sage: A = J0(33)[0]; A._ambient_lattice()
Ambient free module of rank 6 over the principal ideal domain Integer Ring
sage: J0(33)._ambient_lattice()
Ambient free module of rank 6 over the principal ideal domain Integer Ring

_ambient_modular_symbols_abvars( self)

Return a tuple of the ambient modular symbols abelian varieties that make up the Jacobian product that contains self.

Output: tuple of modular symbols abelian varieties

sage: (J0(11) * J0(33))._ambient_modular_symbols_abvars()
(Abelian variety J0(11) of dimension 1, Abelian variety J0(33) of dimension
3)

_ambient_modular_symbols_spaces( self)

Return a tuple of the ambient cuspidal modular symbols spaces that make up the Jacobian product that contains self.

Output: tuple of cuspidal modular symbols spaces

sage: (J0(11) * J0(33))._ambient_modular_symbols_spaces()
(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 6 of Modular Symbols space of
dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field)
sage: (J0(11) * J0(33)[0])._ambient_modular_symbols_spaces()
(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 6 of Modular Symbols space of
dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field)

_ambient_repr( self)

Output: string

sage: (J0(33)*J1(11))._ambient_repr()
'J0(33) x J1(11)'

_classify_ambient_factors( self, [simple=True], [bound=None])

This function implements the following algorithm, which produces data useful in finding a decomposition or complement of self.

  1. Suppose $ A_1 + \cdots + A_n$ is a simple decomposition of the ambient space.
  2. For each $ i$ , let $ B_i = A_1 + \cdots + A_i$ .
  3. For each $ i$ , compute the intersection $ C_i$ of $ B_i$ and self.
  4. For each $ i$ , if the dimension of $ C_i$ is bigger than $ C_{i-1}$ put $ i$ in the ``in'' list; otherwise put $ i$ in the ``out'' list.

Then one can show that self is isogenous to the sum of the $ A_i$ with $ i$ in the ``in'' list. Moreover, the sum of the $ A_j$ with $ i$ in the ``out'' list is a complement of self in the ambient space.

Input:

simple
- bool (default: True)
bound
- integer (default: None); if given, passed onto decomposition function

Output: IN list OUT list simple (or power of simple) factors

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: 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: A = (d1 + d2).image(); A
Abelian subvariety of dimension 1 of J0(33)
sage: A._classify_ambient_factors()
([1], [0, 2], [
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)
])

_complement_shares_no_factors_with_same_label( self)

Return True if no simple factor of self has the same newform_label as any factor in a Poincare complement of self in the ambient product Jacobian.

$ J_0(37)$ is made up of two non-isogenous elliptic curves:

sage: J0(37)[0]._complement_shares_no_factors_with_same_label()
True

$ J_0(33)$ decomposes as a product of two isogenous elliptic curves with a third nonisogenous curve:

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]._complement_shares_no_factors_with_same_label()
False
sage: (D[0]+D[1])._complement_shares_no_factors_with_same_label() 
True
sage: D[2]._complement_shares_no_factors_with_same_label()
True

This example illustrates the relevance of the ambient product Jacobian.

sage: D = (J0(11) * J0(11)).decomposition(); D
[
Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J0(11),
Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J0(11)
]
sage: D[0]._complement_shares_no_factors_with_same_label()
False

This example illustrates that it is the newform label, not the isogeny, class that matters:

sage: D = (J0(11)*J1(11)).decomposition(); D
[
Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J1(11),
Simple abelian subvariety 11aG1(1,11) of dimension 1 of J0(11) x J1(11)
]
sage: D[0]._complement_shares_no_factors_with_same_label()
True
sage: D[0].newform_label()
'11a'
sage: D[1].newform_label()
'11aG1'

_compute_hecke_polynomial( self, n, [var=x])

Return the Hecke polynomial of index $ n$ in terms of the given variable.

Input:

n
- positive integer
var
- string (default: 'x')

sage: A = J0(33)*J0(11)
sage: A._compute_hecke_polynomial(2)
x^4 + 5*x^3 + 6*x^2 - 4*x - 8

_factors_with_same_label( self, other)

Given two modular abelian varieties self and other, this function returns a list of simple abelian subvarieties appearing in the decomposition of self that have the same newform labels. Each simple factor with a given newform label appears at most one.

Input:

other
- abelian variety
Output: list of simple abelian varieties

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]._factors_with_same_label(D[1])
[Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)]
sage: D[0]._factors_with_same_label(D[2])
[]
sage: (D[0]+D[1])._factors_with_same_label(D[1] + D[2])
[Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)]

This illustrates that the multiplicities in the returned list are 1:

sage: (D[0]+D[1])._factors_with_same_label(J0(33))
[Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)]

This illustrates that the ambient product Jacobians do not have to be the same:

sage: (D[0]+D[1])._factors_with_same_label(J0(22))
[Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)]

This illustrates that the actual factor labels are relevant, not just the isogeny class.

sage: (D[0]+D[1])._factors_with_same_label(J1(11))
[]
sage: J1(11)[0].newform_label()
'11aG1'

_Hom_( self, B, [cat=None])

Input:

B
- modular abelian varieties
cat
- category

sage: J0(37)._Hom_(J1(37))
Space of homomorphisms from Abelian variety J0(37) of dimension 2 to
Abelian variety J1(37) of dimension 40

_integral_hecke_matrix( self, n)

Return the matrix of the Hecke operator $ T_n$ acting on the integral homology of this modular abelian variety, if the modular abelian variety is stable under $ T_n$ . Otherwise, raise an ArithmeticError.

sage: A = J0(23)
sage: t = A._integral_hecke_matrix(2); t
[ 0  1 -1  0]
[ 0  1 -1  1]
[-1  2 -2  1]
[-1  1  0 -1]
sage: t.parent()
Full MatrixSpace of 4 by 4 dense matrices over Integer Ring

_isogeny_to_newform_abelian_variety( self)

Return an isogeny from self to an abelian variety $ A_f$ attached to a newform. If self is not simple (so that no such isogeny exists), raise a ValueError.

sage: J0(22)[0]._isogeny_to_newform_abelian_variety()
Abelian variety morphism:
  From: Simple abelian subvariety 11a(1,22) of dimension 1 of J0(22)
  To:   Newform abelian subvariety 11a of dimension 1 of J0(11)
sage: J = J0(11); phi = J.degeneracy_map(33, 1) + J.degeneracy_map(33,3)
sage: A = phi.image()
sage: A._isogeny_to_newform_abelian_variety().matrix()
[-3  3]
[ 0 -3]

_isogeny_to_product_of_powers( self)

Given an abelian variety $ A$ , return an isogeny $ \phi: A \rightarrow B_1 \times \cdots \times B_n$ , where each $ B_i$ is a power of a simple abelian variety. These factors will be exactly those returned by self.decomposition(simple=False).Note that this isogeny is not unique.

sage: J = J0(33) ; D = J.decomposition(simple=False) ; len(D)
2
sage: phi = J._isogeny_to_product_of_powers() ; phi
Abelian variety morphism:
  From: Abelian variety J0(33) of dimension 3
  To:   Abelian subvariety of dimension 3 of J0(33) x J0(33)

sage: J = J0(22) * J0(37)
sage: J._isogeny_to_product_of_powers()
Abelian variety morphism:
  From: Abelian variety J0(22) x J0(37) of dimension 4
  To:   Abelian subvariety of dimension 4 of J0(22) x J0(37) x J0(22) x
J0(37) x J0(22) x J0(37)

_isogeny_to_product_of_simples( self)

Given an abelian variety $ A$ , return an isogeny $ \phi: A \rightarrow B_1 \times \cdots \times B_n$ , where each $ B_i$ is simple. Note that this isogeny is not unique.

sage: J = J0(37) ; J.decomposition()
[
Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37),
Simple abelian subvariety 37b(1,37) of dimension 1 of J0(37)
]
sage: phi = J._isogeny_to_product_of_simples() ; phi
Abelian variety morphism:
  From: Abelian variety J0(37) of dimension 2
  To:   Abelian subvariety of dimension 2 of J0(37) x J0(37)
sage: J[0].intersection(J[1]) == phi.kernel()
True

sage: J = J0(22) * J0(37)
sage: J._isogeny_to_product_of_simples()
Abelian variety morphism:
  From: Abelian variety J0(22) x J0(37) of dimension 4
  To:   Abelian subvariety of dimension 4 of J0(11) x J0(11) x J0(37) x
J0(37)

_quotient_by_abelian_subvariety( self, B)

Return the quotient of self by the abelian variety $ B$ . This is used internally by the quotient and __div__ commmands.

Input:

B
- an abelian subvariety of self
Output:
abelian variety
- quotient $ Q$ of self by B
morphism
- from self to the quotient $ Q$

We compute the new quotient of $ J_0(33)$ .

sage: A = J0(33); B = A.old_subvariety()
sage: Q, f = A._quotient_by_abelian_subvariety(B)

Note that the quotient happens to also be an abelian subvariety:

sage: Q
Abelian subvariety of dimension 1 of J0(33)
sage: Q.lattice()
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: f
Abelian variety morphism:
  From: Abelian variety J0(33) of dimension 3
  To:   Abelian subvariety of dimension 1 of J0(33)

We verify that $ B$ is equal to the kernel of the quotient map.

sage: f.kernel()[1] == B
True

Next we quotient $ J_0(33)$ out by $ Q$ itself:

sage: C, g = A._quotient_by_abelian_subvariety(Q)

The result is not a subvariety:

sage: C
Abelian variety factor of dimension 2 of J0(33)
sage: C.lattice()
Free module of degree 6 and rank 4 over Integer Ring
Echelon basis matrix:
[ 1/3    0    0  2/3   -1    0]
[   0    1    0    0   -1    1]
[   0    0  1/3    0 -2/3  2/3]
[   0    0    0    1   -1   -1]

_quotient_by_finite_subgroup( self, G)

Return the quotient of self by the finite subgroup $ G$ . This is used internally by the quotient and __div__ commmands.

Input:

G
- a finite subgroup of self

Output: abelian variety - the quotient $ Q$ of self by $ G$
morphism
- from self to the quotient $ Q$

We quotient the elliptic curve $ J_0(11)$ out by its cuspidal subgroup.

sage: A = J0(11)
sage: G = A.cuspidal_subgroup(); G
Finite subgroup with invariants [5] over QQ of Abelian variety J0(11) of
dimension 1
sage: Q, f = A._quotient_by_finite_subgroup(G)
sage: Q
Abelian variety factor of dimension 1 of J0(11)
sage: f
Abelian variety morphism:
  From: Abelian variety J0(11) of dimension 1
  To:   Abelian variety factor of dimension 1 of J0(11)

We compute the finite kernel of $ f$ (hence the [0]) and note that it equals the subgroup $ G$ that we quotiented out by:

sage: f.kernel()[0] == G
True

_rational_hecke_matrix( self, n)

Return the matrix of the Hecke operator $ T_n$ acting on the rational homology $ H_1(A,\mathbf{Q})$ of this modular abelian variety, if this action is defined. Otherwise, raise an ArithmeticError.

sage: A = J0(23)
sage: t = A._rational_hecke_matrix(2); t
[ 0  1 -1  0]
[ 0  1 -1  1]
[-1  2 -2  1]
[-1  1  0 -1]
sage: t.parent()
Full MatrixSpace of 4 by 4 dense matrices over Rational Field

_rational_homology_space( self)

Return the rational homology of this modular abelian variety.

sage: J = J0(11)
sage: J._rational_homology_space()
Vector space of dimension 2 over Rational Field

The result is cached:

sage: J._rational_homology_space() is J._rational_homology_space()
True

_repr_( self)

Return string representation of this modular abelian variety.

This is just the generic base class, so it's unlikely to be called in practice.

sage: A = J0(23)
sage: import sage.modular.abvar.abvar as abvar
sage: abvar.ModularAbelianVariety_abstract._repr_(A)
'Abelian variety J0(23) of dimension 2'

sage: (J0(11) * J0(33))._repr_()
'Abelian variety J0(11) x J0(33) of dimension 4'

_simple_isogeny( self, other)

Given self and other, if both are simple, and correspond to the same newform with the same congruence subgroup, return an isogeny. Otherwise, raise a ValueError.

Input:

self, other
- modular abelian varieties
Output: an isogeny

sage: J = J0(33); J
Abelian variety J0(33) of dimension 3
sage: J[0]._simple_isogeny(J[1])
Abelian variety morphism:
  From: Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
  To:   Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33)

The following illustrates how simple isogeny is only implemented when the ambients are the same:

sage: J[0]._simple_isogeny(J1(11))
Traceback (most recent call last):
...
NotImplementedError: _simple_isogeny only implemented when both abelian
variety have the same ambient product Jacobian

Class: ModularAbelianVariety_modsym

class ModularAbelianVariety_modsym
ModularAbelianVariety_modsym( self, modsym, [lattice=None], [newform_level=None], [is_simple=None], [isogeny_number=None], [number=None], [check=True])

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

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

Special Functions: __init__,$ \,$ _modular_symbols

_modular_symbols( self)

Return the modular symbols space that defines this modular abelian variety.

Output: space of modular symbols

sage: M = ModularSymbols(37).cuspidal_submodule()
sage: A = M.abelian_variety(); A
Abelian variety J0(37) of dimension 2
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

Class: ModularAbelianVariety_modsym_abstract

class ModularAbelianVariety_modsym_abstract

Functions: decomposition,$ \,$ dimension,$ \,$ group,$ \,$ groups,$ \,$ is_ambient,$ \,$ is_subvariety,$ \,$ lattice,$ \,$ modular_symbols,$ \,$ new_subvariety,$ \,$ old_subvariety

decomposition( self, [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!

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( self)

Return the dimension of this modular abelian variety.

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

group( self)

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

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( self)

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

Output: tuple

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'>

is_ambient( self)

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

Output: bool

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'>
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( self, other)

Return True if self is a subvariety of other.

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( self)

Return the lattice the defines this modular symbols modular abelian variety.

Output: a free $ \mathbf{Z}$ -module embedded in an ambient $ \mathbf{Q}$ -vector space

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
Echelon basis matrix:
[ 1  0  0 -1  0  0]
[ 0  0  1  0  1 -1]        
sage: type(A)
<class 'sage.modular.abvar.abvar.ModularAbelianVariety_modsym'>

modular_symbols( self, [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.

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( self, [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.

sage: J0(33).new_subvariety()
Abelian subvariety of dimension 1 of J0(33)
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( self, [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.

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)

Special Functions: __add__,$ \,$ _compute_hecke_polynomial,$ \,$ _integral_hecke_matrix,$ \,$ _modular_symbols,$ \,$ _rational_hecke_matrix,$ \,$ _set_lattice

__add__( self, other)

Add two modular abelian variety factors.

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] + D[1]
Abelian subvariety of dimension 2 of J0(42)
sage: D[1].is_subvariety(D[0] + D[1])
True
sage: D[0] + D[1] + D[2]
Abelian subvariety of dimension 3 of J0(42)
sage: D[0] + D[0]
Abelian subvariety of dimension 1 of J0(42)
sage: D[0] + D[0] == D[0]
True
sage: sum(D, D[0]) == A
True

_compute_hecke_polynomial( self, n, [var=x])

Return the characteristic polynomial of the $ n$ -th Hecke operator on self.

NOTE: If self has dimension d, then this is a polynomial of degree d. It is not of degree 2*d, so it is the square root of the characteristic polynomial of the Hecke operator on integral or rational homology (which has degree 2*d).

sage: J0(11).hecke_polynomial(2)
x + 2
sage: J0(23)._compute_hecke_polynomial(2)
x^2 + x - 1
sage: J1(13).hecke_polynomial(2)
x^2 + 3*x + 3
sage: factor(J0(43).hecke_polynomial(2))
(x + 2) * (x^2 - 2)

The Hecke polynomial is the square root of the characteristic polynomial:

sage: factor(J0(43).hecke_operator(2).charpoly())
(x + 2) * (x^2 - 2)

_integral_hecke_matrix( self, n, [sign=0])

Return the action of the Hecke operator $ T_n$ on the integral homology of self.

Input:

n
- a positive integer
sign
- 0, +1, or -1; if 1 or -1 act on the +1 or -1 quotient of the integral homology.

sage: J1(13)._integral_hecke_matrix(2)     # slightly random choice of basis
[-2  0 -1  1]
[ 1 -1  0 -1]
[ 1  1 -2  0]
[ 0  1 -1 -1]
sage: J1(13)._integral_hecke_matrix(2,sign=1)  # slightly random choice of basis
[-1  1]
[-1 -2]
sage: J1(13)._integral_hecke_matrix(2,sign=-1)  # slightly random choice of basis
[-2 -1]
[ 1 -1]

_modular_symbols( self)

Return the space of modular symbols corresponding to this modular symbols abelian variety.

This function is in the abstract base class, so it raises a NotImplementedError:

sage: M = ModularSymbols(37).cuspidal_submodule()
sage: A = M.abelian_variety(); A
Abelian variety J0(37) of dimension 2
sage: sage.modular.abvar.abvar.ModularAbelianVariety_modsym_abstract._modular_symbols(A)
Traceback (most recent call last):
...
NotImplementedError: bug -- must define this

Of course this function isn't called in practice, so this works:

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

_rational_hecke_matrix( self, n, [sign=0])

Return the action of the Hecke operator $ T_n$ on the rational homology of self.

Input:

n
- a positive integer
sign
- 0, +1, or -1; if 1 or -1 act on the +1 or -1 quotient of the rational homology.

sage: J1(13)._rational_hecke_matrix(2)    # slightly random choice of basis
[-2  0 -1  1]
[ 1 -1  0 -1]
[ 1  1 -2  0]
[ 0  1 -1 -1]
sage: J0(43)._rational_hecke_matrix(2,sign=1)  # slightly random choice of basis
[-2  0  1]
[-1 -2  2]
[-2  0  2]

_set_lattice( self, lattice)

Set the lattice of this modular symbols abelian variety.

WARNING: This is only for internal use. Do not use this unless you really really know what you're doing. That's why there is an underscore in this method name.

Input:

lattice
- a lattice

We do something evil - there's no type checking since this function is for internal use only:

sage: A = ModularSymbols(33).cuspidal_submodule().abelian_variety()
sage: A._set_lattice(5)
sage: A.lattice()
5

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