def is_subgroup(self, right):
"""
Return True if self is a subgroup of right.
EXAMPLES::
sage: Gamma1(3).is_subgroup(SL2Z)
True
sage: Gamma1(3).is_subgroup(Gamma1(5))
False
sage: Gamma1(3).is_subgroup(Gamma1(6))
False
sage: Gamma1(6).is_subgroup(Gamma1(3))
True
sage: Gamma1(6).is_subgroup(Gamma0(2))
True
sage: Gamma1(80).is_subgroup(GammaH(40, []))
True
sage: Gamma1(80).is_subgroup(GammaH(40, [21]))
True
"""
if right.level() == 1:
return True
if is_GammaH(right):
return self.level() % right.level() == 0
else:
raise NotImplementedError
def is_subgroup(self, right):
"""
Return True if self is a subgroup of right.
EXAMPLES::
sage: Gamma1(3).is_subgroup(SL2Z)
True
sage: Gamma1(3).is_subgroup(Gamma1(5))
False
sage: Gamma1(3).is_subgroup(Gamma1(6))
False
sage: Gamma1(6).is_subgroup(Gamma1(3))
True
sage: Gamma1(6).is_subgroup(Gamma0(2))
True
sage: Gamma1(80).is_subgroup(GammaH(40, []))
True
sage: Gamma1(80).is_subgroup(GammaH(40, [21]))
True
"""
if right.level() == 1:
return True
if is_GammaH(right):
return self.level() % right.level() == 0
else:
raise NotImplementedError
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 _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 __cmp__(self, other):
"""
Compare self to other.
The ordering on congruence subgroups of the form `\Gamma_H(N)` for
some H is first by level and then by the subgroup H. In
particular, this means that we have `\Gamma_1(N) < \Gamma_H(N) <
\Gamma_0(N)` for every nontrivial subgroup H.
EXAMPLES::
sage: G = Gamma1(86)
sage: G.__cmp__(G)
0
sage: G.__cmp__(GammaH(86, [11])) is not 0
True
sage: Gamma1(12) < Gamma0(12)
True
sage: Gamma1(10) < Gamma1(12)
True
sage: Gamma1(11) == GammaH(11, [])
True
sage: Gamma1(1) == SL2Z
True
sage: Gamma1(2) == Gamma0(2)
True
"""
from all import is_GammaH
if not is_CongruenceSubgroup(other):
return cmp(type(self), type(other))
c = cmp(self.level(), other.level())
if c: return c
# Since Gamma1(N) is GammaH(N) for H all of (Z/N)^\times,
# we know how to compare it to any other GammaH without having
# to look at self._list_of_elements_in_H().
if is_GammaH(other):
if is_Gamma1(other):
return 0
else:
H = other._generators_for_H()
return cmp(0,len(H))
return cmp(type(self), type(other))
def __cmp__(self, other):
"""
Compare self to other.
The ordering on congruence subgroups of the form `\Gamma_H(N)` for
some H is first by level and then by the subgroup H. In
particular, this means that we have `\Gamma_1(N) < \Gamma_H(N) <
\Gamma_0(N)` for every nontrivial subgroup H.
EXAMPLES::
sage: G = Gamma1(86)
sage: G.__cmp__(G)
0
sage: G.__cmp__(GammaH(86, [11])) is not 0
True
sage: Gamma1(12) < Gamma0(12)
True
sage: Gamma1(10) < Gamma1(12)
True
sage: Gamma1(11) == GammaH(11, [])
True
sage: Gamma1(1) == SL2Z
True
sage: Gamma1(2) == Gamma0(2)
True
"""
from all import is_GammaH
if not is_CongruenceSubgroup(other):
return cmp(type(self), type(other))
c = cmp(self.level(), other.level())
if c: return c
# Since Gamma1(N) is GammaH(N) for H all of (Z/N)^\times,
# we know how to compare it to any other GammaH without having
# to look at self._list_of_elements_in_H().
if is_GammaH(other):
if is_Gamma1(other):
return 0
else:
H = other._generators_for_H()
return cmp(0, len(H))
return cmp(type(self), type(other))