def _minimize_level(G):
r"""
Utility function. Given a matrix group `G` contained in `SL(2, \ZZ / N\ZZ)`
for some `N`, test whether or not `G` is the preimage of a subgroup of
smaller level, and if so, return that subgroup.
The trivial group is handled specially: instead of returning a group, it
returns an integer `N`, representing the trivial subgroup of `SL(2, \ZZ /
N\ZZ)`.
EXAMPLE::
sage: M = MatrixSpace(Zmod(9), 2, 2)
sage: G = MatrixGroup([M(x) for x in [[1,1,0,1],[1,3,0,1],[1,0,3,1],[4,0,0,7]]]); G
Matrix group over Ring of integers modulo 9 with 4 generators (
[1 1] [1 3] [1 0] [4 0]
[0 1], [0 1], [3 1], [0 7]
)
sage: sage.modular.arithgroup.congroup_generic._minimize_level(G)
Matrix group over Ring of integers modulo 3 with 1 generators (
[1 1]
[0 1]
)
sage: G = MatrixGroup([M(x) for x in [[1,3,0,1],[1,0,3,1],[4,0,0,7]]]); G
Matrix group over Ring of integers modulo 9 with 3 generators (
[1 3] [1 0] [4 0]
[0 1], [3 1], [0 7]
)
sage: sage.modular.arithgroup.congroup_generic._minimize_level(G)
3
"""
from congroup_gamma import Gamma_constructor as Gamma
Glist = list(G)
N = G.base_ring().characteristic()
i = Gamma(N).index()
for p in N.prime_divisors():
j = Gamma(N // p).index()
k = len(
[g for g in Glist if g.matrix().change_ring(Zmod(N // p)) == 1])
if k == i // j:
if N // p == 1:
return ZZ(1)
H = MatrixGroup(
[g.matrix().change_ring(Zmod(N // p)) for g in G.gens()])
return _minimize_level(H)
# now sanitize the generators (remove duplicates and copies of the identity)
new_gens = [x.matrix() for x in G.gens() if x.matrix() != 1]
all([x.set_immutable() for x in new_gens])
new_gens = list(Set(new_gens))
if new_gens == []:
return ZZ(G.base_ring().characteristic())
return MatrixGroup(new_gens)
def CongruenceSubgroup_constructor(*args):
r"""
Attempt to create a congruence subgroup from the given data.
The allowed inputs are as follows:
- A :class:`~sage.groups.matrix_gps.matrix_group.MatrixGroup` object. This
must be a group of matrices over `\ZZ / N\ZZ` for some `N`, with
determinant 1, in which case the function will return the group of
matrices in `SL(2, \ZZ)` whose reduction mod `N` is in the given group.
- A list of matrices over `\ZZ / N\ZZ` for some `N`. The function will then
compute the subgroup of `SL(2, \ZZ)` generated by these matrices, and
proceed as above.
- An integer `N` and a list of matrices (over any ring coercible to `\ZZ /
N\ZZ`, e.g. over `\ZZ`). The matrices will then be coerced to `\ZZ /
N\ZZ`.
The function checks that the input G is valid. It then tests to see if
`G` is the preimage mod `N` of some group of matrices modulo a proper
divisor `M` of `N`, in which case it replaces `G` with this group before
continuing.
EXAMPLES::
sage: from sage.modular.arithgroup.congroup_generic import CongruenceSubgroup_constructor as CS
sage: CS(2, [[1,1,0,1]])
Congruence subgroup of SL(2,Z) of level 2, preimage of:
Matrix group over Ring of integers modulo 2 with 1 generators (
[1 1]
[0 1]
)
sage: CS([matrix(Zmod(2), 2, [1,1,0,1])])
Congruence subgroup of SL(2,Z) of level 2, preimage of:
Matrix group over Ring of integers modulo 2 with 1 generators (
[1 1]
[0 1]
)
sage: CS(MatrixGroup([matrix(Zmod(2), 2, [1,1,0,1])]))
Congruence subgroup of SL(2,Z) of level 2, preimage of:
Matrix group over Ring of integers modulo 2 with 1 generators (
[1 1]
[0 1]
)
sage: CS(SL(2, 2))
Modular Group SL(2,Z)
Some invalid inputs::
sage: CS(SU(2, 7))
Traceback (most recent call last):
...
TypeError: Ring of definition must be Z / NZ for some N
"""
from sage.groups.matrix_gps.matrix_group import is_MatrixGroup
if is_MatrixGroup(args[0]):
G = args[0]
elif type(args[0]) == type([]):
G = MatrixGroup(args[0])
elif args[0] in ZZ:
M = MatrixSpace(Zmod(args[0]), 2)
G = MatrixGroup([M(x) for x in args[1]])
R = G.matrix_space().base_ring()
if not hasattr(R, "cover_ring") or R.cover_ring() != ZZ:
raise TypeError, "Ring of definition must be Z / NZ for some N"
if not all([x.matrix().det() == 1 for x in G.gens()]):
raise ValueError, "Group must be contained in SL(2, Z / N)"
GG = _minimize_level(G)
if GG in ZZ:
from all import Gamma
return Gamma(GG)
else:
return CongruenceSubgroupFromGroup(GG)