def cusps(self, algorithm='default'):
r"""
Return a sorted list of inequivalent cusps for self, i.e. a set of
representatives for the orbits of self on `\mathbb{P}^1(\QQ)`.
These should be returned in a reduced form where this makes sense.
INPUTS:
- ``algorithm`` -- which algorithm to use to compute the cusps of self.
``'default'`` finds representatives for a known complete set of
cusps. ``'modsym'`` computes the boundary map on the space of weight
two modular symbols associated to self, which finds the cusps for
self in the process.
EXAMPLES::
sage: Gamma0(36).cusps()
[0, 1/18, 1/12, 1/9, 1/6, 1/4, 1/3, 5/12, 1/2, 2/3, 5/6, Infinity]
sage: Gamma0(36).cusps(algorithm='modsym') == Gamma0(36).cusps()
True
sage: GammaH(36, [19,29]).cusps() == Gamma0(36).cusps()
True
sage: Gamma0(1).cusps()
[Infinity]
"""
try:
return copy(self._cusp_list[algorithm])
except (AttributeError, KeyError):
self._cusp_list = {}
from congroup_sl2z import is_SL2Z
if is_SL2Z(self):
from sage.modular.cusps import Cusp
s = [Cusp(1, 0)]
if algorithm == 'default':
s = self._find_cusps()
elif algorithm == 'modsym':
s = sorted(
[self.reduce_cusp(c) for c in self.modular_symbols().cusps()])
else:
raise ValueError, "unknown algorithm: %s" % algorithm
self._cusp_list[algorithm] = s
return copy(s)
def cusps(self, algorithm='default'):
r"""
Return a sorted list of inequivalent cusps for self, i.e. a set of
representatives for the orbits of self on `\mathbb{P}^1(\QQ)`.
These should be returned in a reduced form where this makes sense.
INPUTS:
- ``algorithm`` -- which algorithm to use to compute the cusps of self.
``'default'`` finds representatives for a known complete set of
cusps. ``'modsym'`` computes the boundary map on the space of weight
two modular symbols associated to self, which finds the cusps for
self in the process.
EXAMPLES::
sage: Gamma0(36).cusps()
[0, 1/18, 1/12, 1/9, 1/6, 1/4, 1/3, 5/12, 1/2, 2/3, 5/6, Infinity]
sage: Gamma0(36).cusps(algorithm='modsym') == Gamma0(36).cusps()
True
sage: GammaH(36, [19,29]).cusps() == Gamma0(36).cusps()
True
sage: Gamma0(1).cusps()
[Infinity]
"""
try:
return copy(self._cusp_list[algorithm])
except (AttributeError,KeyError):
self._cusp_list = {}
from congroup_sl2z import is_SL2Z
if is_SL2Z(self):
from sage.modular.cusps import Cusp
s = [Cusp(1,0)]
if algorithm == 'default':
s = self._find_cusps()
elif algorithm == 'modsym':
s = sorted([self.reduce_cusp(c) for c in self.modular_symbols().cusps()])
else:
raise ValueError, "unknown algorithm: %s"%algorithm
self._cusp_list[algorithm] = s
return copy(s)