Congruence Subgroup \Gamma(N)

class sage.modular.arithgroup.congroup_gamma.Gamma_class(level)

Bases: sage.modular.arithgroup.congroup_generic.CongruenceSubgroup

The principal congruence subgroup \Gamma(N).

index()

Return the index of self in the full modular group. This is given by

\prod_{\substack{p \mid N \\ \text{$p$ prime}}}\left(p^{3e}-p^{3e-2}\right).

EXAMPLE::
sage: [Gamma(n).index() for n in [1..19]] [1, 6, 24, 48, 120, 144, 336, 384, 648, 720, 1320, 1152, 2184, 2016, 2880, 3072, 4896, 3888, 6840] sage: Gamma(32041).index() 32893086819240
ncusps()

Return the number of cusps of this subgroup \Gamma(N).

EXAMPLES:

sage: [Gamma(n).ncusps() for n in [1..19]]
[1, 3, 4, 6, 12, 12, 24, 24, 36, 36, 60, 48, 84, 72, 96, 96, 144, 108, 180]
sage: Gamma(30030).ncusps()
278691840
sage: Gamma(2^30).ncusps() 
432345564227567616
sage.modular.arithgroup.congroup_gamma.Gamma_constructor(N)

Return the congruence subgroup \Gamma(N).

EXAMPLES:

sage: Gamma(5) # indirect doctest
Congruence Subgroup Gamma(5)
sage: G = Gamma(23)
sage: G is Gamma(23)
True
sage: G == loads(dumps(G))
True
sage: G is loads(dumps(G))
True
sage.modular.arithgroup.congroup_gamma.is_Gamma(x)

Return True if x is a congruence subgroup of type Gamma.

EXAMPLES:

sage: from sage.modular.arithgroup.all import is_Gamma
sage: is_Gamma(Gamma0(13))
False
sage: is_Gamma(Gamma(4))
True

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