def generators(self, algorithm="farey"):
r"""
Return generators for this congruence subgroup. The result is cached.
INPUT:
- ``algorithm`` (string): either ``farey`` (default) or
``todd-coxeter``.
If ``algorithm`` is set to ``"farey"``, then the generators will be
calculated using Farey symbols, which will always return a *minimal*
generating set. See :mod:`~sage.modular.arithgroup.farey_symbol` for
more information.
If ``algorithm`` is set to ``"todd-coxeter"``, a simpler algorithm
based on Todd-Coxeter enumeration will be used. This tends to return
far larger sets of generators.
EXAMPLES::
sage: GammaH(7, [2]).generators()
[
[1 1] [ 2 -1] [ 4 -3]
[0 1], [ 7 -3], [ 7 -5]
]
sage: GammaH(7, [2]).generators(algorithm="todd-coxeter")
[
[1 1] [-90 29] [ 15 4] [-10 -3] [ 1 -1] [1 0] [1 1] [-3 -1]
[0 1], [301 -97], [-49 -13], [ 7 2], [ 0 1], [7 1], [0 1], [ 7 2],
<BLANKLINE>
[-13 4] [-5 -1] [-5 -2] [-10 3] [ 1 0] [ 9 -1] [-20 7]
[ 42 -13], [21 4], [28 11], [ 63 -19], [-7 1], [28 -3], [-63 22],
<BLANKLINE>
[1 0] [-3 -1] [ 15 -4] [ 2 -1] [ 22 -7] [-5 1] [ 8 -3]
[7 1], [ 7 2], [ 49 -13], [ 7 -3], [ 63 -20], [14 -3], [-21 8],
<BLANKLINE>
[11 5] [-13 -4]
[35 16], [-42 -13]
]
"""
if algorithm == "farey":
return self.farey_symbol().generators()
elif algorithm == "todd-coxeter":
from sage.modular.modsym.ghlist import GHlist
from .congroup import generators_helper
level = self.level()
gen_list = generators_helper(GHlist(self), level)
return [self(g, check=False) for g in gen_list]
else:
raise ValueError(
"Unknown algorithm '%s' (should be either 'farey' or 'todd-coxeter')"
% algorithm)
def generators(self):
r"""
Return generators for this congruence subgroup.
The result is cached.
EXAMPLE::
sage: for g in GammaH(3, [2]).generators():
... print g
... print '---'
[1 1]
[0 1]
---
[-1 0]
[ 0 -1]
---
[ 1 -1]
[ 0 1]
---
[1 0]
[3 1]
---
[1 1]
[0 1]
---
[-1 0]
[ 3 -1]
---
[ 1 0]
[-3 1]
---
"""
from sage.modular.modsym.ghlist import GHlist
from congroup_pyx import generators_helper
level = self.level()
gen_list = generators_helper(GHlist(self), level, Mat2Z)
return [self(g, check=False) for g in gen_list]