def _new_group_from_level(self, level):
r"""
Return a new group of the same type (Gamma0, Gamma1, or
GammaH) as self of the given level. In the case that self is of type
GammaH, we take the largest H inside `(\ZZ/ \text{level}\ZZ)^\times`
which maps to H, namely its inverse image under the natural reduction
map.
EXAMPLES::
sage: G = Gamma0(20)
sage: G._new_group_from_level(4)
Congruence Subgroup Gamma0(4)
sage: G._new_group_from_level(40)
Congruence Subgroup Gamma0(40)
sage: G = Gamma1(10)
sage: G._new_group_from_level(6)
Traceback (most recent call last):
...
ValueError: one level must divide the other
sage: G = GammaH(50,[7]); G
Congruence Subgroup Gamma_H(50) with H generated by [7]
sage: G._new_group_from_level(25)
Congruence Subgroup Gamma_H(25) with H generated by [7]
sage: G._new_group_from_level(10)
Congruence Subgroup Gamma0(10)
sage: G._new_group_from_level(100)
Congruence Subgroup Gamma_H(100) with H generated by [7, 57]
"""
from congroup_gamma0 import is_Gamma0
from congroup_gamma1 import is_Gamma1
from congroup_gammaH import is_GammaH
from all import Gamma0, Gamma1, GammaH
N = self.level()
if (level % N) and (N % level):
raise ValueError, "one level must divide the other"
if is_Gamma0(self):
return Gamma0(level)
elif is_Gamma1(self):
return Gamma1(level)
elif is_GammaH(self):
H = self._generators_for_H()
if level > N:
d = level // N
diffs = [N * i for i in range(d)]
newH = [h + diff for h in H for diff in diffs]
return GammaH(level, [x for x in newH if gcd(level, x) == 1])
else:
return GammaH(level, [h % level for h in H])
else:
raise NotImplementedError
def gamma_h_subgroups(self):
r"""
Return the subgroups of the form `\Gamma_H(N)` contained
in self, where `N` is the level of self.
EXAMPLES::
sage: G = Gamma0(11)
sage: G.gamma_h_subgroups()
[Congruence Subgroup Gamma0(11), Congruence Subgroup Gamma_H(11) with H generated by [4], Congruence Subgroup Gamma_H(11) with H generated by [10], Congruence Subgroup Gamma1(11)]
sage: G = Gamma0(12)
sage: G.gamma_h_subgroups()
[Congruence Subgroup Gamma0(12), Congruence Subgroup Gamma_H(12) with H generated by [7], Congruence Subgroup Gamma_H(12) with H generated by [11], Congruence Subgroup Gamma_H(12) with H generated by [5], Congruence Subgroup Gamma1(12)]
"""
from all import GammaH
N = self.level()
R = IntegerModRing(N)
return [GammaH(N, H) for H in R.multiplicative_subgroups()]