# Ambient Jacobian abelian variety#

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
```

```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)#

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),)
```
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: J0(81).newform_decomposition('a')
[q - 2*q^4 + O(q^6), q - 2*q^4 + O(q^6), q + a0*q^2 + q^4 - a0*q^5 + O(q^6)]

sage: J1(19).newform_decomposition('a')
[q - 2*q^3 - 2*q^4 + 3*q^5 + O(q^6),
q + a1*q^2 + (-1/9*a1^5 - 1/3*a1^4 - 1/3*a1^3 + 1/3*a1^2 - a1 - 1)*q^3 + (4/9*a1^5 + 2*a1^4 + 14/3*a1^3 + 17/3*a1^2 + 6*a1 + 2)*q^4 + (-2/3*a1^5 - 11/3*a1^4 - 10*a1^3 - 14*a1^2 - 15*a1 - 9)*q^5 + O(q^6)]
```