Congruence arithmetic subgroups of {\rm SL}_2(\ZZ)

Sage can compute extensively with the standard congruence subgroups \Gamma_0(N), \Gamma_1(N), and \Gamma_H(N).

AUTHORS:

  • William Stein
  • David Loeffler (2009, 10) – modifications to work with more general arithmetic subgroups
class sage.modular.arithgroup.congroup_generic.CongruenceSubgroup(level)

Bases: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup

is_congruence()

Return True, since this is a congruence subgroup.

EXAMPLE:

sage: Gamma0(7).is_congruence()
True
level()

Return the level of this congruence subgroup.

EXAMPLES:

sage: SL2Z.level()
1
sage: Gamma0(20).level()
20
sage: Gamma1(11).level()
11
sage: GammaH(14, [5]).level()
14
modular_abelian_variety()

Return the modular abelian variety corresponding to the congruence subgroup self.

EXAMPLES:

sage: Gamma0(11).modular_abelian_variety()
Abelian variety J0(11) of dimension 1
sage: Gamma1(11).modular_abelian_variety()
Abelian variety J1(11) of dimension 1
sage: GammaH(11,[3]).modular_abelian_variety()
Abelian variety JH(11,[3]) of dimension 1
modular_symbols(sign=0, weight=2, base_ring=Rational Field)

Return the space of modular symbols of the specified weight and sign on the congruence subgroup self.

EXAMPLES:

sage: G = Gamma0(23)
sage: G.modular_symbols()
Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Rational Field
sage: G.modular_symbols(weight=4)
Modular Symbols space of dimension 12 for Gamma_0(23) of weight 4 with sign 0 over Rational Field
sage: G.modular_symbols(base_ring=GF(7))
Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Finite Field of size 7
sage: G.modular_symbols(sign=1)
Modular Symbols space of dimension 3 for Gamma_0(23) of weight 2 with sign 1 over Rational Field
sage.modular.arithgroup.congroup_generic.is_CongruenceSubgroup(x)

Return True if x is of type CongruenceSubgroup.

EXAMPLES:

sage: from sage.modular.arithgroup.congroup_generic import is_CongruenceSubgroup
sage: is_CongruenceSubgroup(SL2Z)
True
sage: is_CongruenceSubgroup(Gamma0(13))
True
sage: is_CongruenceSubgroup(Gamma1(6))
True
sage: is_CongruenceSubgroup(GammaH(11, [3]))
True
sage: is_CongruenceSubgroup(SymmetricGroup(3))
False

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Elements of Arithmetic Subgroups

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Congruence Subgroup \Gamma_H(N)

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