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# Ambient Jacobian Abelian Variety

TESTS:

    sage: loads(dumps(J0(37))) == J0(37)
    True
    sage: loads(dumps(J1(13))) == J1(13)
    True

`sage.modular.abvar.abvar_ambient_jacobian.``ModAbVar_ambient_jacobian`(group)

  Return the ambient Jacobian attached to a given congruence subgroup.

  The result is cached using a weakref. This function is called internally by modular abelian variety constructors.

  INPUT:

  - `group` - a congruence subgroup.

  OUTPUT: a modular abelian variety attached

  EXAMPLES:

      sage: import sage.modular.abvar.abvar_ambient_jacobian as abvar_ambient_jacobian
      sage: A = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(11))
      sage: A
      Abelian variety J0(11) of dimension 1
      sage: B = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(11))
      sage: A is B
      True

  You can get access to and/or clear the cache as follows:

      sage: abvar_ambient_jacobian._cache = {}
      sage: B = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(11))
      sage: A is B
      False

class `sage.modular.abvar.abvar_ambient_jacobian.``ModAbVar_ambient_jacobian_class`(group)

  Bases: `sage.modular.abvar.abvar.ModularAbelianVariety_modsym_abstract`

  An ambient Jacobian modular abelian variety attached to a congruence subgroup.

  `ambient_variety`()

    Return the ambient modular abelian variety that contains self. Since self is a Jacobian modular abelian variety, this is just self.

    OUTPUT: abelian variety

    EXAMPLES:

        sage: A = J0(17)
        sage: A.ambient_variety()
        Abelian variety J0(17) of dimension 1
        sage: A is A.ambient_variety()
        True

  `decomposition`(simple=True, bound=None)

    Decompose this ambient Jacobian as a product of abelian subvarieties, up to isogeny.

    EXAMPLES:

        sage: J0(33).decomposition(simple=False)
        [
        Abelian subvariety of dimension 2 of J0(33),
        Abelian subvariety of dimension 1 of J0(33)
        ]
        sage: J0(33).decomposition(simple=False)[1].is_simple()
        True
        sage: J0(33).decomposition(simple=False)[0].is_simple()
        False
        sage: J0(33).decomposition(simple=False)
        [
        Abelian subvariety of dimension 2 of J0(33),
        Simple abelian subvariety 33a(None,33) of dimension 1 of J0(33)
        ]
        sage: J0(33).decomposition(simple=True)
        [
        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)
        ]

  `degeneracy_map`(level, t=1, check=True)

    Return the t-th degeneracy map from self to J(level). Here t must be a divisor of either level/self.level() or self.level()/level.

    INPUT:

    - `level` - integer (multiple or divisor of level of self)
    - `t` - divisor of quotient of level of self and level
    - `check` - bool (default: True); if True do some checks on the input

    OUTPUT: a morphism

    EXAMPLES:

        sage: J0(11).degeneracy_map(33)
        Degeneracy map from Abelian variety J0(11) of dimension 1 to Abelian variety J0(33) of dimension 3 defined by [1]
        sage: J0(11).degeneracy_map(33).matrix()
        [ 0 -3  2  1 -2  0]
        [ 1 -2  0  1  0 -1]
        sage: J0(11).degeneracy_map(33,3).matrix()
        [-1  0  0  0  1 -2]
        [-1 -1  1 -1  1  0]
        sage: J0(33).degeneracy_map(11,1).matrix()
        [ 0  1]
        [ 0 -1]
        [ 1 -1]
        [ 0  1]
        [-1  1]
        [ 0  0]
        sage: J0(11).degeneracy_map(33,1).matrix() * J0(33).degeneracy_map(11,1).matrix()
        [4 0]
        [0 4]

  `dimension`()

    Return the dimension of this modular abelian variety.

    EXAMPLES:

        sage: J0(2007).dimension()
        221
        sage: J1(13).dimension()
        2
        sage: J1(997).dimension()
        40920
        sage: J0(389).dimension()
        32
        sage: JH(389,[4]).dimension()
        64
        sage: J1(389).dimension()
        6112

  `group`()

    Return the group that this Jacobian modular abelian variety is attached to.

    EXAMPLES:

        sage: J1(37).group()
        Congruence Subgroup Gamma1(37)
        sage: J0(5077).group()
        Congruence Subgroup Gamma0(5077)
        sage: J = GammaH(11,[3]).modular_abelian_variety(); J
        Abelian variety JH(11,[3]) of dimension 1
        sage: J.group()
        Congruence Subgroup Gamma_H(11) with H generated by [3]

  `groups`()

    Return the tuple of congruence subgroups attached to this ambient Jacobian. This is always a tuple of length 1.

    OUTPUT: tuple

    EXAMPLES:

        sage: J0(37).groups()
        (Congruence Subgroup Gamma0(37),)

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