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Package sage :: Package modular :: Package abvar :: Module abvar_ambient_jacobian :: Class ModAbVar_ambient_jacobian_class
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Class ModAbVar_ambient_jacobian_class

source code

                      object --+                    
                               |                    
structure.sage_object.SageObject --+                
                                   |                
             structure.parent.Parent --+            
                                       |            
    structure.parent_base.ParentWithBase --+        
                                           |        
        abvar.ModularAbelianVariety_abstract --+    
                                               |    
     abvar.ModularAbelianVariety_modsym_abstract --+
                                                   |
                                                  ModAbVar_ambient_jacobian_class


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


Instance Methods [hide private]
 
__init__(self, group)
Create an ambient Jacobian modular abelian variety.		 source code
 
_modular_symbols(self)
Return the modular symbols space associated to this ambient Jacobian.		 source code
 
_repr_(self)
Return string representation of this Jacobian modular abelian variety.		 source code
 
_latex_(self)
Return Latex representation of self.		 source code
 
ambient_variety(self)
Return the ambient modular abelian variety that contains self.		 source code
 
group(self)
Return the group that this Jacobian modular abelian variety is attached to.		 source code
 
groups(self)
Return the tuple of congruence subgroups attached to this ambient Jacobian.		 source code
 
_calculate_endomorphism_generators(self)
Calculate generators for the endomorphism ring of self.		 source code
 
degeneracy_map(self, level, t=1, check=True)
Return the t-th degeneracy map from self to J(level).		 source code
 
dimension(self)
Return the dimension of this modular abelian variety.		 source code
 
decomposition(self, simple=True, bound=['4ti2-20061025', 'R-2.6.0', 'atlas-3.7.37', 'atlas-3.8.1', 'a...)
Decompose this ambient Jacobian as a product of abelian subvarieties, up to isogeny.		 source code

Inherited from abvar.ModularAbelianVariety_modsym_abstract: __add__, is_ambient, is_subvariety, lattice, modular_symbols, new_subvariety, old_subvariety

Inherited from abvar.ModularAbelianVariety_modsym_abstract (private): _compute_hecke_polynomial, _integral_hecke_matrix, _rational_hecke_matrix, _set_lattice

Inherited from abvar.ModularAbelianVariety_abstract: _Hom_, __cmp__, __contains__, __div__, __getitem__, __getslice__, __mul__, __pow__, __radd__, ambient_morphism, base_extend, base_field, category, change_ring, complement, cuspidal_subgroup, degen_t, degree, direct_product, dual, endomorphism_ring, finite_subgroup, free_module, hecke_operator, hecke_polynomial, homology, in_same_ambient_variety, integral_homology, intersection, is_hecke_stable, is_simple, is_subvariety_of_ambient_jacobian, isogeny_number, label, 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

Inherited from abvar.ModularAbelianVariety_abstract (private): _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, _factors_with_same_label, _isogeny_to_newform_abelian_variety, _isogeny_to_product_of_powers, _isogeny_to_product_of_simples, _quotient_by_abelian_subvariety, _quotient_by_finite_subgroup, _rational_homology_space, _simple_isogeny

Inherited from structure.parent_base.ParentWithBase: Hom, __new__, base, base_extend_canonical, base_extend_canonical_sym, base_extend_recursive, base_ring

Inherited from structure.parent.Parent: _coerce_, coerce_map_from, coerce_map_from_impl, construction, get_action, get_action_impl, has_coerce_map_from, has_coerce_map_from_impl, init_coerce

Inherited from structure.parent.Parent (private): _an_element, _an_element_impl, _coerce_impl, _coerce_self, _coerce_try

Inherited from structure.sage_object.SageObject: __hash__, __repr__, _axiom_, _axiom_init_, _gap_, _gap_init_, _gp_, _gp_init_, _interface_, _interface_init_, _interface_is_cached_, _kash_, _kash_init_, _macaulay2_, _macaulay2_init_, _magma_, _magma_init_, _maple_, _maple_init_, _mathematica_, _mathematica_init_, _maxima_, _maxima_init_, _octave_, _octave_init_, _pari_, _pari_init_, _r_init_, _sage_, _singular_, _singular_init_, db, dump, dumps, plot, rename, reset_name, save, version

Inherited from object: __delattr__, __getattribute__, __reduce__, __reduce_ex__, __setattr__, __str__

Properties [hide private]

Inherited from structure.parent.Parent (private): _has_coerce_map_from

Inherited from object: __class__

Method Details [hide private]

__init__(self, group)
(Constructor)

source code  

Create an ambient Jacobian modular abelian variety.

EXAMPLES:
    sage: A = J0(37); A
    Abelian variety J0(37) of dimension 2
    sage: type(A)
    <class 'sage.modular.abvar.abvar_ambient_jacobian.ModAbVar_ambient_jacobian_class'>
    sage: A.group()
    Congruence Subgroup Gamma0(37)
Overrides: abvar.ModularAbelianVariety_abstract.__init__

_modular_symbols(self)

source code  

Return the modular symbols space associated to this ambient Jacobian.

OUTPUT:
    modular symbols space

EXAMPLES:
    sage: M = J0(33)._modular_symbols(); M
    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(33)._modular_symbols() is M
    True
Overrides: abvar.ModularAbelianVariety_modsym_abstract._modular_symbols

_repr_(self)

source code  

Return string representation of this Jacobian modular abelian
variety.

EXAMPLES:
    sage: A = J0(11); A
    Abelian variety J0(11) of dimension 1
    sage: A._repr_()
    'Abelian variety J0(11) of dimension 1'
    sage: A.rename("J_0(11)")
    sage: A
    J_0(11)

We now clear the cache to get rid of our renamed $J_0(11)$.
    sage: import sage.modular.abvar.abvar_ambient_jacobian as abvar_ambient_jacobian
    sage: abvar_ambient_jacobian._cache = {}
Overrides: abvar.ModularAbelianVariety_abstract._repr_

_latex_(self)

source code  

Return Latex representation of self.

EXAMPLES:
    sage: latex(J0(37))
    J_0(37)
    sage: J1(13)._latex_()
    'J_1(13)'
    sage: latex(JH(389,[2]))
    J_H(389,[2])

ambient_variety(self)

source code  

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
Overrides: abvar.ModularAbelianVariety_abstract.ambient_variety

group(self)

source code  

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]
Overrides: abvar.ModularAbelianVariety_modsym_abstract.group

groups(self)

source code  

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),)
Overrides: abvar.ModularAbelianVariety_modsym_abstract.groups

_calculate_endomorphism_generators(self)

source code  

Calculate generators for the endomorphism ring of
self.

EXAMPLES:
    sage: J0(11)._calculate_endomorphism_generators()
    [Abelian variety endomorphism of Abelian variety J0(11) of dimension 1]
    sage: ls = J0(46)._calculate_endomorphism_generators() ; ls
    [Abelian variety endomorphism of Abelian variety J0(46) of dimension 5,
     Abelian variety endomorphism of Abelian variety J0(46) of dimension 5,
     Abelian variety endomorphism of Abelian variety J0(46) of dimension 5,
     Abelian variety endomorphism of Abelian variety J0(46) of dimension 5,
     Abelian variety endomorphism of Abelian variety J0(46) of dimension 5]
    sage: len(ls) == J0(46).dimension()
    True

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

source code  

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]
Overrides: abvar.ModularAbelianVariety_abstract.degeneracy_map

dimension(self)

source code  

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
Overrides: abvar.ModularAbelianVariety_modsym_abstract.dimension

decomposition(self, simple=True, bound=['4ti2-20061025', 'R-2.6.0', 'atlas-3.7.37', 'atlas-3.8.1', 'a...)

source code  

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)
    ]
Overrides: abvar.ModularAbelianVariety_modsym_abstract.decomposition

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