def nu2(self):
r"""
Return the number of orbits of elliptic points of order 2 for this
arithmetic subgroup.
EXAMPLES::
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().nu2()
Traceback (most recent call last):
...
NotImplementedError
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nu2(Gamma0(1105)) == 8
True
"""
# Subgroups not containing -1 have no elliptic points of order 2.
if not self.is_even():
return 0
# Cheap trick: if self is a subgroup of something with no elliptic points,
# then self has no elliptic points either.
from all import Gamma0, is_CongruenceSubgroup
if is_CongruenceSubgroup(self):
if self.is_subgroup(Gamma0(self.level())) and Gamma0(
self.level()).nu2() == 0:
return 0
# Otherwise, the number of elliptic points is the number of g in self \
# SL2Z such that the stabiliser of g * i in self is not trivial. (Note
# that the points g*i for g in the coset reps are not distinct, but it
# still works, since the failure of these points to be distinct happens
# precisely when the preimages are not elliptic.)
from all import SL2Z
count = 0
for g in self.coset_reps():
if g * SL2Z([0, 1, -1, 0]) * (~g) in self:
count += 1
return count
def nu2(self):
r"""
Return the number of orbits of elliptic points of order 2 for this
arithmetic subgroup.
EXAMPLES::
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().nu2()
Traceback (most recent call last):
...
NotImplementedError
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nu2(Gamma0(1105)) == 8
True
"""
# Subgroups not containing -1 have no elliptic points of order 2.
if not self.is_even():
return 0
# Cheap trick: if self is a subgroup of something with no elliptic points,
# then self has no elliptic points either.
from all import Gamma0, is_CongruenceSubgroup
if is_CongruenceSubgroup(self):
if self.is_subgroup(Gamma0(self.level())) and Gamma0(self.level()).nu2() == 0:
return 0
# Otherwise, the number of elliptic points is the number of g in self \
# SL2Z such that the stabiliser of g * i in self is not trivial. (Note
# that the points g*i for g in the coset reps are not distinct, but it
# still works, since the failure of these points to be distinct happens
# precisely when the preimages are not elliptic.)
from all import SL2Z
count = 0
for g in self.coset_reps():
if g * SL2Z([0,1,-1,0]) * (~g) in self:
count += 1
return count
def nu3(self):
r"""
Return the number of orbits of elliptic points of order 3 for this
arithmetic subgroup.
EXAMPLES::
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().nu3()
Traceback (most recent call last):
...
NotImplementedError
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nu3(Gamma0(1729)) == 8
True
We test that a bug in handling of subgroups not containing -1 is fixed: ::
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nu3(GammaH(7, [2]))
2
"""
# Cheap trick: if self is a subgroup of something with no elliptic points,
# then self has no elliptic points either.
from all import Gamma0, is_CongruenceSubgroup
if is_CongruenceSubgroup(self):
if self.is_subgroup(Gamma0(self.level())) and Gamma0(
self.level()).nu3() == 0:
return 0
from all import SL2Z
count = 0
for g in self.coset_reps():
if g * SL2Z([0, 1, -1, -1]) * (~g) in self:
count += 1
if self.is_even():
return count
else:
return count / 2
def nu3(self):
r"""
Return the number of orbits of elliptic points of order 3 for this
arithmetic subgroup.
EXAMPLES::
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().nu3()
Traceback (most recent call last):
...
NotImplementedError
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nu3(Gamma0(1729)) == 8
True
We test that a bug in handling of subgroups not containing -1 is fixed: ::
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nu3(GammaH(7, [2]))
2
"""
# Cheap trick: if self is a subgroup of something with no elliptic points,
# then self has no elliptic points either.
from all import Gamma0, is_CongruenceSubgroup
if is_CongruenceSubgroup(self):
if self.is_subgroup(Gamma0(self.level())) and Gamma0(self.level()).nu3() == 0:
return 0
from all import SL2Z
count = 0
for g in self.coset_reps():
if g * SL2Z([0,1,-1,-1]) * (~g) in self:
count += 1
if self.is_even():
return count
else:
return count/2