Congruence Subgroup \Gamma(N)

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

The principal congruence subgroup \Gamma(N).

__call__(x, check=True)

Create an element of this congruence subgroup from x.

If the optional flag check is True (default), check whether x actually gives an element of self.

EXAMPLES:

sage: G = Gamma(5)
sage: G([1, 0, -10, 1])
[ 1   0]
[-10  1]
sage: G(matrix(ZZ, 2, [26, 5, 5, 1]))
[26  5]
[ 5  1]
sage: G([1, 1, 6, 7])
...
TypeError: matrix must have diagonal entries (=1, 7) congruent to 1
modulo 5, and off-diagonal entries (=1,6) divisible by 5
__cmp__(other)

Compare self to other.

EXAMPLES:

sage: Gamma(3) == SymmetricGroup(8)
False
sage: Gamma(3) == Gamma1(3)
False
sage: Gamma(5) < Gamma(6)
True
sage: Gamma(5) == Gamma(5)
True
__reduce__()

Used for pickling self.

EXAMPLES:

sage: Gamma(5).__reduce__()
(<function Gamma_constructor at ...>, (5,))
_repr_()

Return the string representation of self.

EXAMPLES:

sage: Gamma(133)._repr_()
'Congruence Subgroup Gamma(133)'
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
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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