def GammaH_constructor(level, H):
r"""
Return the congruence subgroup `\Gamma_H(N)`, which is the subgroup of
`SL_2(\ZZ)` consisting of matrices of the form `\begin{pmatrix} a & b \\
c & d \end{pmatrix}` with `N | c` and `a, b \in H`, for `H` a specified
subgroup of `(\ZZ/N\ZZ)^\times`.
INPUT:
- level -- an integer
- H -- either 0, 1, or a list
* If H is a list, return `\Gamma_H(N)`, where `H`
is the subgroup of `(\ZZ/N\ZZ)^*` **generated** by the
elements of the list.
* If H = 0, returns `\Gamma_0(N)`.
* If H = 1, returns `\Gamma_1(N)`.
EXAMPLES::
sage: GammaH(11,0) # indirect doctest
Congruence Subgroup Gamma0(11)
sage: GammaH(11,1)
Congruence Subgroup Gamma1(11)
sage: GammaH(11,[10])
Congruence Subgroup Gamma_H(11) with H generated by [10]
sage: GammaH(11,[10,1])
Congruence Subgroup Gamma_H(11) with H generated by [10]
sage: GammaH(14,[10])
Traceback (most recent call last):
...
ArithmeticError: The generators [10] must be units modulo 14
"""
from all import Gamma0, Gamma1, SL2Z
if level == 1:
return SL2Z
elif H == 0:
return Gamma0(level)
elif H == 1:
return Gamma1(level)
H = _normalize_H(H, level)
if H == []:
return Gamma1(level)
Hlist = _list_subgroup(level, H)
if len(Hlist) == euler_phi(level):
return Gamma0(level)
key = (level, tuple(H))
try:
return _gammaH_cache[key]
except KeyError:
_gammaH_cache[key] = GammaH_class(level, H, Hlist)
return _gammaH_cache[key]
def index(self):
r"""
Return the index of self in SL2Z.
EXAMPLE::
sage: [G.index() for G in Gamma0(40).gamma_h_subgroups()]
[72, 144, 144, 144, 144, 288, 288, 288, 288, 144, 288, 288, 576, 576, 144, 288, 288, 576, 576, 144, 288, 288, 576, 576, 288, 576, 1152]
"""
from all import Gamma1
return Gamma1(self.level()).index() / len(self._list_of_elements_in_H())
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