def generators(self):
r"""
Return generators for this congruence subgroup.
The result is cached.
EXAMPLE::
sage: for g in Gamma0(3).generators():
... print g
... print '---'
[1 1]
[0 1]
---
[-1 0]
[ 0 -1]
---
...
---
[ 1 0]
[-3 1]
---
"""
from sage.modular.modsym.p1list import P1List
from congroup_pyx import generators_helper
level = self.level()
if level == 1: # P1List isn't very happy working mod 1
return [ self([0,-1,1,0]), self([1,1,0,1]) ]
gen_list = generators_helper(P1List(level), level, Mat2Z)
return [self(g, check=False) for g in gen_list]
def generators(self):
r"""
Return generators for this congruence subgroup.
The result is cached.
EXAMPLE::
sage: for g in Gamma1(3).generators():
... print g
... print '---'
[1 1]
[0 1]
---
[-20 9]
[ 51 -23]
---
[ 4 1]
[-9 -2]
---
...
---
[ 4 -1]
[ 9 -2]
---
[ -5 3]
[-12 7]
---
"""
from sage.modular.modsym.g1list import G1list
from congroup_pyx import generators_helper
level = self.level()
gen_list = generators_helper(G1list(level), level, Mat2Z)
return [self(g, check=False) for g in gen_list]
def generators(self, algorithm="farey"):
r"""
Return generators for this congruence subgroup.
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.
EXAMPLE::
sage: Gamma0(3).generators()
[
[1 1] [-1 1]
[0 1], [-3 2]
]
sage: Gamma0(3).generators(algorithm="todd-coxeter")
[
[1 1] [-1 0] [ 1 -1] [1 0] [1 1] [-1 0] [ 1 0]
[0 1], [ 0 -1], [ 0 1], [3 1], [0 1], [ 3 -1], [-3 1]
]
sage: SL2Z.gens()
(
[ 0 -1] [1 1]
[ 1 0], [0 1]
)
"""
if self.level() == 1:
# we return a fixed set of generators for SL2Z, for historical
# reasons, which aren't the ones the Farey symbol code gives
return [ self([0,-1,1,0]), self([1,1,0,1]) ]
elif algorithm=="farey":
return self.farey_symbol().generators()
elif algorithm=="todd-coxeter":
from sage.modular.modsym.p1list import P1List
from congroup_pyx import generators_helper
level = self.level()
if level == 1: # P1List isn't very happy working mod 1
return [ self([0,-1,1,0]), self([1,1,0,1]) ]
gen_list = generators_helper(P1List(level), level, Mat2Z)
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, algorithm="farey"):
r"""
Return generators for this congruence subgroup.
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.
EXAMPLE::
sage: Gamma0(3).generators()
[
[1 1] [-1 1]
[0 1], [-3 2]
]
sage: Gamma0(3).generators(algorithm="todd-coxeter")
[
[1 1] [-1 0] [ 1 -1] [1 0] [1 1] [-1 0] [ 1 0]
[0 1], [ 0 -1], [ 0 1], [3 1], [0 1], [ 3 -1], [-3 1]
]
sage: SL2Z.gens()
(
[ 0 -1] [1 1]
[ 1 0], [0 1]
)
"""
if self.level() == 1:
# we return a fixed set of generators for SL2Z, for historical
# reasons, which aren't the ones the Farey symbol code gives
return [self([0, -1, 1, 0]), self([1, 1, 0, 1])]
elif algorithm == "farey":
return self.farey_symbol().generators()
elif algorithm == "todd-coxeter":
from sage.modular.modsym.p1list import P1List
from congroup_pyx import generators_helper
level = self.level()
if level == 1: # P1List isn't very happy working mod 1
return [self([0, -1, 1, 0]), self([1, 1, 0, 1])]
gen_list = generators_helper(P1List(level), level, Mat2Z)
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, 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.
EXAMPLE::
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_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]
else:
raise ValueError(
"Unknown algorithm '%s' (should be either 'farey' or 'todd-coxeter')"
% algorithm)
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.
EXAMPLE::
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_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]
else:
raise ValueError, "Unknown algorithm '%s' (should be either 'farey' or 'todd-coxeter')" % algorithm
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.
EXAMPLE::
sage: Gamma1(3).generators()
[
[1 1] [ 1 -1]
[0 1], [ 3 -2]
]
sage: Gamma1(3).generators(algorithm="todd-coxeter")
[
[1 1] [-20 9] [ 4 1] [-5 -2] [ 1 -1] [1 0] [1 1] [-5 2]
[0 1], [ 51 -23], [-9 -2], [ 3 1], [ 0 1], [3 1], [0 1], [12 -5],
<BLANKLINE>
[ 1 0] [ 4 -1] [ -5 3] [ 1 -1] [ 7 -3] [ 4 -1] [ -5 3]
[-3 1], [ 9 -2], [-12 7], [ 3 -2], [12 -5], [ 9 -2], [-12 7]
]
"""
if algorithm == "farey":
return self.farey_symbol().generators()
elif algorithm == "todd-coxeter":
from sage.modular.modsym.g1list import G1list
from congroup_pyx import generators_helper
level = self.level()
gen_list = generators_helper(G1list(level), level, Mat2Z)
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, 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.
EXAMPLE::
sage: Gamma1(3).generators()
[
[1 1] [ 1 -1]
[0 1], [ 3 -2]
]
sage: Gamma1(3).generators(algorithm="todd-coxeter")
[
[1 1] [-20 9] [ 4 1] [-5 -2] [ 1 -1] [1 0] [1 1] [-5 2]
[0 1], [ 51 -23], [-9 -2], [ 3 1], [ 0 1], [3 1], [0 1], [12 -5],
<BLANKLINE>
[ 1 0] [ 4 -1] [ -5 3] [ 1 -1] [ 7 -3] [ 4 -1] [ -5 3]
[-3 1], [ 9 -2], [-12 7], [ 3 -2], [12 -5], [ 9 -2], [-12 7]
]
"""
if algorithm=="farey":
return self.farey_symbol().generators()
elif algorithm=="todd-coxeter":
from sage.modular.modsym.g1list import G1list
from congroup_pyx import generators_helper
level = self.level()
gen_list = generators_helper(G1list(level), level, Mat2Z)
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]
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]