def __init__(self, level):
r"""
The congruence subgroup `\Gamma_0(N)`.
EXAMPLES::
sage: G = Gamma0(11); G
Congruence Subgroup Gamma0(11)
sage: TestSuite(G).run()
sage: G is loads(dumps(G))
True
TESTS::
sage: g = Gamma0(5)([1,1,0,1])
sage: g in Gamma0(7)
True
sage: g = Gamma0(5)([1,0,5,1])
sage: g in Gamma0(7)
False
sage: g = Gamma0(2)([1,0,0,1])
sage: g in SL2Z
True
"""
CongruenceSubgroup.__init__(self, level)
def __init__(self, level):
r"""
The congruence subgroup `\Gamma_0(N)`.
EXAMPLES::
sage: G = Gamma0(11); G
Congruence Subgroup Gamma0(11)
sage: TestSuite(G).run()
sage: G is loads(dumps(G))
True
TESTS::
sage: g = Gamma0(5)([1,1,0,1])
sage: g in Gamma0(7)
True
sage: g = Gamma0(5)([1,0,5,1])
sage: g in Gamma0(7)
False
sage: g = Gamma0(2)([1,0,0,1])
sage: g in SL2Z
True
"""
CongruenceSubgroup.__init__(self, level)
def __cmp__(self, other):
"""
Compare self to other.
The ordering on congruence subgroups of the form GammaH(N) for
some H is first by level and then by the subgroup H. In
particular, this means that we have Gamma1(N) < GammaH(N) <
Gamma0(N) for every nontrivial subgroup H.
EXAMPLES::
sage: G = GammaH(86, [9])
sage: G.__cmp__(G)
0
sage: G.__cmp__(GammaH(86, [11])) is not 0
True
sage: Gamma1(11) < Gamma0(11)
True
sage: Gamma1(11) == GammaH(11, [])
True
sage: Gamma0(11) == GammaH(11, [2])
True
"""
if is_GammaH(other):
t = cmp(self.level(), other.level())
if t:
return t
else:
return cmp(self._list_of_elements_in_H(), other._list_of_elements_in_H())
else:
return CongruenceSubgroup.__cmp__(self, other)
def __cmp__(self, other):
"""
Compare self to other.
The ordering on congruence subgroups of the form GammaH(N) for
some H is first by level and then by the subgroup H. In
particular, this means that we have Gamma1(N) < GammaH(N) <
Gamma0(N) for every nontrivial subgroup H.
EXAMPLES::
sage: G = GammaH(86, [9])
sage: G.__cmp__(G)
0
sage: G.__cmp__(GammaH(86, [11])) is not 0
True
sage: Gamma1(11) < Gamma0(11)
True
sage: Gamma1(11) == GammaH(11, [])
True
sage: Gamma0(11) == GammaH(11, [2])
True
"""
if is_GammaH(other):
t = cmp(self.level(), other.level())
if t:
return t
else:
return cmp(self._list_of_elements_in_H(),
other._list_of_elements_in_H())
else:
return CongruenceSubgroup.__cmp__(self, other)
def __cmp__(self, other):
"""
Compare self to other.
The ordering on congruence subgroups of the form GammaH(N) for some H
is first by level, then by the order of H, then lexicographically by H.
In particular, this means that we have Gamma1(N) < GammaH(N) <
Gamma0(N) for every nontrivial proper subgroup H.
EXAMPLES::
sage: G = GammaH(86, [9])
sage: G.__cmp__(G)
0
sage: G.__cmp__(GammaH(86, [11])) is not 0
True
sage: Gamma1(11) < Gamma0(11)
True
sage: Gamma1(11) == GammaH(11, [])
True
sage: Gamma0(11) == GammaH(11, [2])
True
sage: G = Gamma0(86)
sage: G.__cmp__(G)
0
sage: G.__cmp__(GammaH(86, [11])) is not 0
True
sage: Gamma1(17) < Gamma0(17)
True
sage: Gamma0(1) == SL2Z
True
sage: Gamma0(2) == Gamma1(2)
True
sage: [x._list_of_elements_in_H() for x in sorted(Gamma0(24).gamma_h_subgroups())]
[[1],
[1, 5],
[1, 7],
[1, 11],
[1, 13],
[1, 17],
[1, 19],
[1, 23],
[1, 5, 7, 11],
[1, 5, 13, 17],
[1, 5, 19, 23],
[1, 7, 13, 19],
[1, 7, 17, 23],
[1, 11, 13, 23],
[1, 11, 17, 19],
[1, 5, 7, 11, 13, 17, 19, 23]]
"""
if is_GammaH(other):
return (cmp(self.level(), other.level())
or -cmp(self.index(), other.index())
or cmp(self._list_of_elements_in_H(),
other._list_of_elements_in_H()))
else:
return CongruenceSubgroup.__cmp__(self, other)
def __cmp__(self, other):
"""
Compare self to other.
The ordering on congruence subgroups of the form GammaH(N) for some H
is first by level, then by the order of H, then lexicographically by H.
In particular, this means that we have Gamma1(N) < GammaH(N) <
Gamma0(N) for every nontrivial proper subgroup H.
EXAMPLES::
sage: G = GammaH(86, [9])
sage: G.__cmp__(G)
0
sage: G.__cmp__(GammaH(86, [11])) is not 0
True
sage: Gamma1(11) < Gamma0(11)
True
sage: Gamma1(11) == GammaH(11, [])
True
sage: Gamma0(11) == GammaH(11, [2])
True
sage: G = Gamma0(86)
sage: G.__cmp__(G)
0
sage: G.__cmp__(GammaH(86, [11])) is not 0
True
sage: Gamma1(17) < Gamma0(17)
True
sage: Gamma0(1) == SL2Z
True
sage: Gamma0(2) == Gamma1(2)
True
sage: [x._list_of_elements_in_H() for x in sorted(Gamma0(24).gamma_h_subgroups())]
[[1],
[1, 5],
[1, 7],
[1, 11],
[1, 13],
[1, 17],
[1, 19],
[1, 23],
[1, 5, 7, 11],
[1, 5, 13, 17],
[1, 5, 19, 23],
[1, 7, 13, 19],
[1, 7, 17, 23],
[1, 11, 13, 23],
[1, 11, 17, 19],
[1, 5, 7, 11, 13, 17, 19, 23]]
"""
if is_GammaH(other):
return (cmp(self.level(), other.level())
or -cmp(self.index(), other.index())
or cmp(self._list_of_elements_in_H(), other._list_of_elements_in_H()))
else:
return CongruenceSubgroup.__cmp__(self, other)
def __init__(self, level, H, Hlist=None):
r"""
The congruence subgroup `\Gamma_H(N)`. The subgroup H
must be input as a list.
EXAMPLES::
sage: GammaH(117, [4])
Congruence Subgroup Gamma_H(117) with H generated by [4]
sage: G = GammaH(16, [7])
sage: TestSuite(G).run()
sage: G is loads(dumps(G))
True
"""
CongruenceSubgroup.__init__(self, level)
self.__H = H
if Hlist is None: Hlist = _list_subgroup(level, H)
self.__Hlist = Hlist
def are_equivalent(self, x, y, trans=False):
r"""
Check if the cusps `x` and `y` are equivalent under the action of this group.
ALGORITHM: The cusps `u_1 / v_1` and `u_2 / v_2` are equivalent modulo
`\Gamma(N)` if and only if `(u_1, v_1) = \pm (u_2, v_2) \bmod N`.
EXAMPLE::
sage: Gamma(7).are_equivalent(Cusp(2/3), Cusp(5/4))
True
"""
if trans:
return CongruenceSubgroup.are_equivalent(self, x,y,trans=trans)
N = self.level()
u1,v1 = (x.numerator() % N, x.denominator() % N)
u2,v2 = (y.numerator(), y.denominator())
return ((u1,v1) == (u2 % N, v2 % N)) or ((u1,v1) == (-u2 % N, -v2 % N))
def are_equivalent(self, x, y, trans=False):
r"""
Check if the cusps `x` and `y` are equivalent under the action of this group.
ALGORITHM: The cusps `u_1 / v_1` and `u_2 / v_2` are equivalent modulo
`\Gamma(N)` if and only if `(u_1, v_1) = \pm (u_2, v_2) \bmod N`.
EXAMPLE::
sage: Gamma(7).are_equivalent(Cusp(2/3), Cusp(5/4))
True
"""
if trans:
return CongruenceSubgroup.are_equivalent(self, x,y,trans=trans)
N = self.level()
u1,v1 = (x.numerator() % N, x.denominator() % N)
u2,v2 = (y.numerator(), y.denominator())
return ((u1,v1) == (u2 % N, v2 % N)) or ((u1,v1) == (-u2 % N, -v2 % N))
def __cmp__(self, other):
r"""
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
sage: Gamma(3) == Gamma(3).as_permutation_group()
True
"""
if is_Gamma(other):
return cmp(self.level(), other.level())
else:
return CongruenceSubgroup.__cmp__(self, other)
def __cmp__(self, other):
r"""
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
sage: Gamma(3) == Gamma(3).as_permutation_group()
True
"""
if is_Gamma(other):
return cmp(self.level(), other.level())
else:
return CongruenceSubgroup.__cmp__(self, other)
