def is_discriminant(self, D, primitive=True):
r"""
Returns whether ``D`` is a discriminant of an element of ``self``.
Note: Checking that something isn't a discriminant takes much
longer than checking for valid discriminants.
INPUT:
- ``D`` -- An element of the base ring.
- ``primitive`` -- If ``True`` (default) then only primitive
elements are considered.
OUPUT:
``True`` if ``D`` is a primitive discriminant (a discriminant of
a primitive element) and ``False`` otherwise.
If ``primitive=False`` then also non-primitive elements are considered.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=4)
sage: G.is_discriminant(68)
True
sage: G.is_discriminant(196, primitive=False) # long time
True
sage: G.is_discriminant(2)
False
"""
self._conjugacy_representatives(0)
t_bound = max(coerce_AA(0), coerce_AA(
(D + 4) / (self.lam()**2))).sqrt().floor()
for t in range(self._max_block_length + 1, t_bound + 1):
self._conjugacy_representatives(t)
if D in self._conj_prim:
return True
if not primitive and D in self._conj_nonprim:
return True
if D in self._conj_prim:
return True
elif not primitive and D in self._conj_nonprim:
return True
else:
return False
def is_discriminant(self, D, primitive=True):
r"""
Returns whether ``D`` is a discriminant of an element of ``self``.
Note: Checking that something isn't a discriminant takes much
longer than checking for valid discriminants.
INPUT:
- ``D`` -- An element of the base ring.
- ``primitive`` -- If ``True`` (default) then only primitive
elements are considered.
OUTPUT:
``True`` if ``D`` is a primitive discriminant (a discriminant of
a primitive element) and ``False`` otherwise.
If ``primitive=False`` then also non-primitive elements are considered.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=4)
sage: G.is_discriminant(68)
True
sage: G.is_discriminant(196, primitive=False) # long time
True
sage: G.is_discriminant(2)
False
"""
self._conjugacy_representatives(0)
t_bound = max(coerce_AA(0), coerce_AA((D + 4)/(self.lam()**2))).sqrt().floor()
for t in range(self._max_block_length + 1, t_bound + 1):
self._conjugacy_representatives(t)
if D in self._conj_prim:
return True
if not primitive and D in self._conj_nonprim:
return True
if D in self._conj_prim:
return True
elif not primitive and D in self._conj_nonprim:
return True
else:
return False
def rho(self):
r"""
Return the vertex ``rho`` of the basic hyperbolic
triangle which describes ``self``. ``rho`` has
absolute value 1 and angle ``pi/n``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(3).rho() == 1/2 + sqrt(3)/2*i
True
sage: HeckeTriangleGroup(4).rho() == sqrt(2)/2*(1 + i)
True
sage: HeckeTriangleGroup(6).rho() == sqrt(3)/2 + 1/2*i
True
sage: HeckeTriangleGroup(10).rho()
0.95105651629515...? + 0.30901699437494...?*I
sage: HeckeTriangleGroup(infinity).rho()
1
"""
# TODO: maybe rho should be replaced by -rhobar
# Also we could use NumberFields...
if (self._n == infinity):
return coerce_AA(1)
else:
rho = AlgebraicField()(exp(pi / self._n * i))
rho.simplify()
return rho
def rho(self):
r"""
Return the vertex ``rho`` of the basic hyperbolic
triangle which describes ``self``. ``rho`` has
absolute value 1 and angle ``pi/n``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(3).rho() == 1/2 + sqrt(3)/2*i
True
sage: HeckeTriangleGroup(4).rho() == sqrt(2)/2*(1 + i)
True
sage: HeckeTriangleGroup(6).rho() == sqrt(3)/2 + 1/2*i
True
sage: HeckeTriangleGroup(10).rho()
0.95105651629515...? + 0.30901699437494...?*I
sage: HeckeTriangleGroup(infinity).rho()
1
"""
# TODO: maybe rho should be replaced by -rhobar
# Also we could use NumberFields...
if (self._n == infinity):
return coerce_AA(1)
else:
rho = AlgebraicField()(exp(pi/self._n*i))
rho.simplify()
return rho
def list_discriminants(self, D, primitive=True, hyperbolic=True, incomplete=False):
r"""
Returns a list of all discriminants up to some upper bound ``D``.
INPUT:
- ``D`` -- An element/discriminant of the base ring or
more generally an upper bound for the discriminant.
- ``primitive`` -- If ``True`` (default) then only primitive
discriminants are listed.
- ``hyperbolic`` -- If ``True`` (default) then only positive
discriminants are listed.
- ``incomplete`` -- If ``True`` (default: ``False``) then all (also higher)
discriminants which were gathered so far are listed
(however there might be missing discriminants inbetween).
OUTPUT:
A list of discriminants less than or equal to ``D``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=4)
sage: G.list_discriminants(D=68)
[4, 12, 14, 28, 32, 46, 60, 68]
sage: G.list_discriminants(D=0, hyperbolic=False, primitive=False)
[-4, -2, 0]
sage: G = HeckeTriangleGroup(n=5)
sage: G.list_discriminants(D=20)
[4*lam, 7*lam + 6, 9*lam + 5]
sage: G.list_discriminants(D=0, hyperbolic=False, primitive=False)
[-4, -lam - 2, lam - 3, 0]
"""
self._conjugacy_representatives(D=D)
if incomplete:
max_D = infinity
else:
max_D = coerce_AA(D)
L = []
if hyperbolic:
L += [key for key in self._conj_prim if coerce_AA(key) > 0 and coerce_AA(key) <= max_D]
else:
L += [key for key in self._conj_prim if coerce_AA(key) <= max_D]
if not primitive:
if hyperbolic:
L += [key for key in self._conj_nonprim if coerce_AA(key) > 0 and coerce_AA(key) <= max_D and key not in L]
else:
L += [key for key in self._conj_nonprim if coerce_AA(key) <= max_D and key not in L]
return sorted(L, key=coerce_AA)
def class_number(self, D, primitive=True):
r"""
Return the class number of ``self`` for the discriminant ``D``.
I.e. the number of conjugacy classes of (primitive) elements
of discriminant ``D``.
Note: Due to the 1-1 correspondence with hyperbolic fixed
points resp. hyperbolic binary quadratic forms
this also gives the class number in those cases.
INPUT:
- ``D`` -- An element of the base ring corresponding
to a valid discriminant.
- ``primitive`` -- If ``True`` (default) then only primitive
elements are considered.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=4)
sage: G.class_number(4)
1
sage: G.class_number(4, primitive=False)
1
sage: G.class_number(14)
2
sage: G.class_number(32)
2
sage: G.class_number(32, primitive=False)
3
sage: G.class_number(68)
4
"""
if coerce_AA(D) <= 0:
raise NotImplementedError
self._conjugacy_representatives(D=D)
num = ZZ(0)
if D in self._conj_prim:
num = len(self._conj_prim[D])
if not primitive and D in self._conj_nonprim:
num += len(self._conj_nonprim[D])
if num == 0:
raise ValueError("D = {} is not a{} discriminant for {}".format(
D, " primitive" if primitive else "", self))
else:
return num
def class_number(self, D, primitive=True):
r"""
Return the class number of ``self`` for the discriminant ``D``.
I.e. the number of conjugacy classes of (primitive) elements
of discriminant ``D``.
Note: Due to the 1-1 correspondence with hyperbolic fixed
points resp. hyperbolic binary quadratic forms
this also gives the class number in those cases.
INPUT:
- ``D`` -- An element of the base ring corresponding
to a valid discriminant.
- ``primitive`` -- If ``True`` (default) then only primitive
elements are considered.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=4)
sage: G.class_number(4)
1
sage: G.class_number(4, primitive=False)
1
sage: G.class_number(14)
2
sage: G.class_number(32)
2
sage: G.class_number(32, primitive=False)
3
sage: G.class_number(68)
4
"""
if coerce_AA(D) <= 0:
raise NotImplementedError
self._conjugacy_representatives(D=D)
num = ZZ(0)
if D in self._conj_prim:
num = len(self._conj_prim[D])
if not primitive and D in self._conj_nonprim:
num += len(self._conj_nonprim[D])
if num == 0:
raise ValueError("D = {} is not a{} discriminant for {}".format(D, " primitive" if primitive else "", self))
else:
return num
def lam_minpoly(self):
r"""
Return the minimal polynomial of the corresponding lambda parameter of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(10).lam_minpoly()
x^4 - 5*x^2 + 5
sage: HeckeTriangleGroup(17).lam_minpoly()
x^8 - x^7 - 7*x^6 + 6*x^5 + 15*x^4 - 10*x^3 - 10*x^2 + 4*x + 1
sage: HeckeTriangleGroup(infinity).lam_minpoly()
x - 2
"""
# TODO: Write an explicit (faster) implementation
lam_symbolic = 2 * cos(pi / self._n)
return coerce_AA(lam_symbolic).minpoly()
def lam_minpoly(self):
r"""
Return the minimal polynomial of the corresponding lambda parameter of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(10).lam_minpoly()
x^4 - 5*x^2 + 5
sage: HeckeTriangleGroup(17).lam_minpoly()
x^8 - x^7 - 7*x^6 + 6*x^5 + 15*x^4 - 10*x^3 - 10*x^2 + 4*x + 1
sage: HeckeTriangleGroup(infinity).lam_minpoly()
x - 2
"""
# TODO: Write an explicit (faster) implementation
lam_symbolic = 2*cos(pi/self._n)
return coerce_AA(lam_symbolic).minpoly()
def __init__(self, n):
r"""
Hecke triangle group (2, n, infinity).
Namely the von Dyck group corresponding to the triangle group
with angles (pi/2, pi/n, 0).
INPUT:
- ``n`` - ``infinity`` or an integer greater or equal to ``3``.
OUTPUT:
The Hecke triangle group for the given parameter ``n``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(12)
sage: G
Hecke triangle group for n = 12
sage: G.category()
Category of groups
"""
self._n = n
self.element_repr_method("default")
if n in [3, infinity]:
self._base_ring = ZZ
self._lam = ZZ(1) if n == 3 else ZZ(2)
else:
lam_symbolic = 2 * cos(pi / n)
K = NumberField(self.lam_minpoly(),
'lam',
embedding=coerce_AA(lam_symbolic))
#self._base_ring = K.order(K.gens())
self._base_ring = K.maximal_order()
self._lam = self._base_ring.gen(1)
T = matrix(self._base_ring, [[1, self._lam], [0, 1]])
S = matrix(self._base_ring, [[0, -1], [1, 0]])
FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(2),
self._base_ring, [S, T])
def __init__(self, n):
r"""
Hecke triangle group (2, n, infinity).
Namely the von Dyck group corresponding to the triangle group
with angles (pi/2, pi/n, 0).
INPUT:
- ``n`` - ``infinity`` or an integer greater or equal to ``3``.
OUTPUT:
The Hecke triangle group for the given parameter ``n``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(12)
sage: G
Hecke triangle group for n = 12
sage: G.category()
Category of groups
"""
self._n = n
self.element_repr_method("default")
if n in [3, infinity]:
self._base_ring = ZZ
self._lam = ZZ(1) if n==3 else ZZ(2)
else:
lam_symbolic = 2*cos(pi/n)
K = NumberField(self.lam_minpoly(), 'lam', embedding = coerce_AA(lam_symbolic))
#self._base_ring = K.order(K.gens())
self._base_ring = K.maximal_order()
self._lam = self._base_ring.gen(1)
T = matrix(self._base_ring, [[1,self._lam],[0,1]])
S = matrix(self._base_ring, [[0,-1],[1,0]])
FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(2), self._base_ring, [S, T])
def list_discriminants(self,
D,
primitive=True,
hyperbolic=True,
incomplete=False):
r"""
Returns a list of all discriminants up to some upper bound ``D``.
INPUT:
- ``D`` -- An element/discriminant of the base ring or
more generally an upper bound for the discriminant.
- ``primitive`` -- If ``True`` (default) then only primitive
discriminants are listed.
- ``hyperbolic`` -- If ``True`` (default) then only positive
discriminants are listed.
- ``incomplete`` -- If ``True`` (default: ``False``) then all (also higher)
discriminants which were gathered so far are listed
(however there might be missing discriminants inbetween).
OUPUT:
A list of discriminants less than or equal to ``D``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=4)
sage: G.list_discriminants(D=68)
[4, 12, 14, 28, 32, 46, 60, 68]
sage: G.list_discriminants(D=0, hyperbolic=False, primitive=False)
[-4, -2, 0]
sage: G = HeckeTriangleGroup(n=5)
sage: G.list_discriminants(D=20)
[4*lam, 7*lam + 6, 9*lam + 5]
sage: G.list_discriminants(D=0, hyperbolic=False, primitive=False)
[-4, -lam - 2, lam - 3, 0]
"""
self._conjugacy_representatives(D=D)
if incomplete:
max_D = infinity
else:
max_D = coerce_AA(D)
L = []
if hyperbolic:
L += [
key for key in self._conj_prim
if coerce_AA(key) > 0 and coerce_AA(key) <= max_D
]
else:
L += [key for key in self._conj_prim if coerce_AA(key) <= max_D]
if not primitive:
if hyperbolic:
L += [
key for key in self._conj_nonprim if coerce_AA(key) > 0
and coerce_AA(key) <= max_D and key not in L
]
else:
L += [
key for key in self._conj_nonprim
if coerce_AA(key) <= max_D and key not in L
]
return sorted(L, key=coerce_AA)
def _conjugacy_representatives(self, max_block_length=ZZ(0), D=None):
r"""
Store conjugacy representatives up to block length
``max_block_length`` (a non-negative integer, default: 0)
in the internal dictionary. Previously calculated data is reused.
This is a helper function for e.g. :meth:`class_number`.
The set of all (hyperbolic) conjugacy types of block length
``t`` is stored in ``self._conj_block[t]``.
The set of all primitive representatives (so far) with
discriminant ``D`` is stored in ``self._conj_prim[D]``.
The set of all non-primitive representatives (so far) with
discriminant ``D`` is stored in ``self._conj_nonprim[D]``.
The case of non-positive discriminants is done manually.
INPUT:
- ``max_block_length`` -- A non-negative integer (default: ``0``),
the maximal block length.
- ``D`` -- An element/discriminant of the base ring or
more generally an upper bound for the
involved discriminants. If ``D != None``
then an upper bound for ``max_block_length``
is deduced from ``D`` (default: ``None``).
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=5)
sage: G.element_repr_method("conj")
sage: G._conjugacy_representatives(2)
sage: list(G._conj_block[2])
[((4, 1), (3, 1)), ((2, 2),), ((3, 2),), ((3, 1), (1, 1)), ((4, 1), (1, 1)), ((4, 1), (2, 1)), ((3, 1), (2, 1)), ((2, 1), (1, 1))]
sage: [key for key in sorted(G._conj_prim)]
[-4, lam - 3, 0, 4*lam, 7*lam + 6, 9*lam + 5, 15*lam + 6, 33*lam + 21]
sage: for key in sorted(G._conj_prim):
....: print G._conj_prim[key]
[[S], [S]]
[[U], [U]]
[[V(4)]]
[[V(3)], [V(2)]]
[[V(1)*V(4)]]
[[V(3)*V(4)], [V(1)*V(2)]]
[[V(1)*V(3)], [V(2)*V(4)]]
[[V(2)*V(3)]]
sage: [key for key in sorted(G._conj_nonprim)]
[-lam - 2, lam - 3, 32*lam + 16]
sage: for key in sorted(G._conj_nonprim):
....: print G._conj_nonprim[key]
[[U^(-2)], [U^2], [U^(-2)], [U^2]]
[[U^(-1)], [U^(-1)]]
[[V(2)^2], [V(3)^2]]
sage: G.element_repr_method("default")
"""
from sage.combinat.partition import OrderedPartitions
from sage.combinat.combinat import tuples
from sage.arith.all import divisors
if not D is None:
max_block_length = max(coerce_AA(0), coerce_AA((D + 4)/(self.lam()**2))).sqrt().floor()
else:
try:
max_block_length = ZZ(max_block_length)
if max_block_length < 0:
raise TypeError
except TypeError:
raise ValueError("max_block_length must be a non-negative integer!")
if not hasattr(self, "_max_block_length"):
self._max_block_length = ZZ(0)
self._conj_block = {}
self._conj_nonprim = {}
self._conj_prim = {}
# It is not clear how to define the class number for D=0:
# Conjugacy classes are V(n-1)^(+-k) for arbitrary k
# and the trivial class (what about self_conj_block[0]?).
#
# One way is to define it using the fixed points and in
# that case V(n-1) would be a good representative.
# The non-primitive case is unclear however...
#
# We set it here to ensure that 0 is enlisted as a discriminant...
#
self._conj_prim[ZZ(0)] = []
self._conj_prim[ZZ(0)].append(self.V(self.n()-1))
self._elliptic_conj_reps()
if max_block_length <= self._max_block_length:
return
def is_cycle(seq):
length = len(seq)
for n in divisors(length):
if n < length and is_cycle_of_length(seq, n):
return True
return False
def is_cycle_of_length(seq, n):
for i in range(n, len(seq)):
if seq[i] != seq[i % n]:
return False
return True
j_list = range(1, self.n())
for t in range(self._max_block_length + 1, max_block_length + 1):
t_ZZ = ZZ(t)
if t_ZZ not in self._conj_block:
self._conj_block[t_ZZ] = set()
partitions = OrderedPartitions(t).list()
for par in partitions:
len_par = len(par)
exp_list = tuples(j_list, len_par)
for ex in exp_list:
keep = True
if len_par > 1:
for k in range(-1,len_par-1):
if ex[k] == ex[k+1]:
keep = False
break
# We don't want powers of V(1)
elif ex[0] == 1:
keep = False
# But: Do we maybe want powers of V(n-1)??
# For now we exclude the parabolic cases...
elif ex[0] == self.n()-1:
keep = False
if keep:
conj_type = cyclic_representative(tuple((ZZ(ex[k]), ZZ(par[k])) for k in range(len_par)))
self._conj_block[t_ZZ].add(conj_type)
for el in self._conj_block[t_ZZ]:
group_el = prod([self.V(el[k][0])**el[k][1] for k in range(len(el))])
#if el != group_el.conjugacy_type():
# raise AssertionError("This shouldn't happen!")
D = group_el.discriminant()
if coerce_AA(D) < 0:
raise AssertionError("This shouldn't happen!")
if coerce_AA(D) == 0:
raise AssertionError("This shouldn't happen!")
#continue
# The primitive cases
#if group_el.is_primitive():
if not ((len(el) == 1 and el[0][1] > 1) or is_cycle(el)):
if D not in self._conj_prim:
self._conj_prim[D] = []
self._conj_prim[D].append(group_el)
# The remaining cases
else:
if D not in self._conj_nonprim:
self._conj_nonprim[D] = []
self._conj_nonprim[D].append(group_el)
self._max_block_length = max_block_length
def root_extension_embedding(self, D, K=None):
r"""
Return the correct embedding from the root extension field
of the given discriminant ``D`` to the field ``K``.
Also see the method ``root_extension_embedding(K)`` of
``HeckeTriangleGroupElement`` for more examples.
INPUT:
- ``D`` -- An element of the base ring of ``self``
corresponding to a discriminant.
- ``K`` -- A field to which we want the (correct) embeddding.
If ``K=None`` (default) then ``AlgebraicField()`` is
used for positive ``D`` and ``AlgebraicRealField()``
otherwise.
OUTPUT:
The corresponding embedding if it was found.
Otherwise a ValueError is raised.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=infinity)
sage: G.root_extension_embedding(32)
Ring morphism:
From: Number Field in e with defining polynomial x^2 - 32
To: Algebraic Real Field
Defn: e |--> 5.656854249492...?
sage: G.root_extension_embedding(-4)
Ring morphism:
From: Number Field in e with defining polynomial x^2 + 4
To: Algebraic Field
Defn: e |--> 2*I
sage: G.root_extension_embedding(4)
Ring Coercion morphism:
From: Rational Field
To: Algebraic Real Field
sage: G = HeckeTriangleGroup(n=7)
sage: lam = G.lam()
sage: D = 4*lam^2 + 4*lam - 4
sage: G.root_extension_embedding(D, CC)
Relative number field morphism:
From: Number Field in e with defining polynomial x^2 - 4*lam^2 - 4*lam + 4 over its base field
To: Complex Field with 53 bits of precision
Defn: e |--> 4.02438434522...
lam |--> 1.80193773580...
sage: D = lam^2 - 4
sage: G.root_extension_embedding(D)
Relative number field morphism:
From: Number Field in e with defining polynomial x^2 - lam^2 + 4 over its base field
To: Algebraic Field
Defn: e |--> 0.?... + 0.867767478235...?*I
lam |--> 1.801937735804...?
"""
D = self.base_ring()(D)
F = self.root_extension_field(D)
if K is None:
if coerce_AA(D) > 0:
K = AA
else:
K = AlgebraicField()
L = [emb for emb in F.embeddings(K)]
# Three possibilities up to numerical artefacts:
# (1) emb = e, purely imaginary
# (2) emb = e or lam (can't distinguish), purely real
# (3) emb = (e,lam), e purely imaginary, lam purely real
# (4) emb = (e,lam), e purely real, lam purely real
# There always exists one emb with "e" positive resp. positive imaginary
# and if there is a lam there exists a positive one...
#
# Criteria to pick the correct "maximum":
# 1. First figure out if e resp. lam is purely real or imaginary
# (using "abs(e.imag()) > abs(e.real())")
# 2. In the purely imaginary case we don't want anything negative imaginary
# and we know the positive case is unique after sorting lam
# 3. For the remaining cases we want the biggest real part
# (and lam should get comparison priority)
def emb_key(emb):
L = []
gens_len = len(emb.im_gens())
for k in range(gens_len):
a = emb.im_gens()[k]
try:
a.simplify()
a.exactify()
except AttributeError:
pass
# If a is purely imaginary:
if abs(a.imag()) > abs(a.real()):
if a.imag() < 0:
a = -infinity
else:
a = ZZ(0)
else:
a = a.real()
L.append(a)
L.reverse()
return L
if len(L) > 1:
L.sort(key = emb_key)
return L[-1]
def root_extension_embedding(self, D, K=None):
r"""
Return the correct embedding from the root extension field
of the given discriminant ``D`` to the field ``K``.
Also see the method ``root_extension_embedding(K)`` of
``HeckeTriangleGroupElement`` for more examples.
INPUT:
- ``D`` -- An element of the base ring of ``self``
corresponding to a discriminant.
- ``K`` -- A field to which we want the (correct) embeddding.
If ``K=None`` (default) then ``AlgebraicField()`` is
used for positive ``D`` and ``AlgebraicRealField()``
otherwise.
OUTPUT:
The corresponding embedding if it was found.
Otherwise a ValueError is raised.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=infinity)
sage: G.root_extension_embedding(32)
Ring morphism:
From: Number Field in e with defining polynomial x^2 - 32
To: Algebraic Real Field
Defn: e |--> 5.656854249492...?
sage: G.root_extension_embedding(-4)
Ring morphism:
From: Number Field in e with defining polynomial x^2 + 4
To: Algebraic Field
Defn: e |--> 2*I
sage: G.root_extension_embedding(4)
Ring Coercion morphism:
From: Rational Field
To: Algebraic Real Field
sage: G = HeckeTriangleGroup(n=7)
sage: lam = G.lam()
sage: D = 4*lam^2 + 4*lam - 4
sage: G.root_extension_embedding(D, CC)
Relative number field morphism:
From: Number Field in e with defining polynomial x^2 - 4*lam^2 - 4*lam + 4 over its base field
To: Complex Field with 53 bits of precision
Defn: e |--> 4.02438434522...
lam |--> 1.80193773580...
sage: D = lam^2 - 4
sage: G.root_extension_embedding(D)
Relative number field morphism:
From: Number Field in e with defining polynomial x^2 - lam^2 + 4 over its base field
To: Algebraic Field
Defn: e |--> 0.?... + 0.867767478235...?*I
lam |--> 1.801937735804...?
"""
D = self.base_ring()(D)
F = self.root_extension_field(D)
if K is None:
if coerce_AA(D) > 0:
K = AA
else:
K = AlgebraicField()
L = [emb for emb in F.embeddings(K)]
# Three possibilities up to numerical artefacts:
# (1) emb = e, purely imaginary
# (2) emb = e or lam (can't distinguish), purely real
# (3) emb = (e,lam), e purely imaginary, lam purely real
# (4) emb = (e,lam), e purely real, lam purely real
# There always exists one emb with "e" positive resp. positive imaginary
# and if there is a lam there exists a positive one...
#
# Criteria to pick the correct "maximum":
# 1. First figure out if e resp. lam is purely real or imaginary
# (using "abs(e.imag()) > abs(e.real())")
# 2. In the purely imaginary case we don't want anything negative imaginary
# and we know the positive case is unique after sorting lam
# 3. For the remaining cases we want the biggest real part
# (and lam should get comparison priority)
def emb_key(emb):
L = []
gens_len = len(emb.im_gens())
for k in range(gens_len):
a = emb.im_gens()[k]
try:
a.simplify()
a.exactify()
except AttributeError:
pass
# If a is purely imaginary:
if abs(a.imag()) > abs(a.real()):
if a.imag() < 0:
a = -infinity
else:
a = ZZ(0)
else:
a = a.real()
L.append(a)
L.reverse()
return L
if len(L) > 1:
L.sort(key=emb_key)
return L[-1]
def class_representatives(self, D, primitive=True):
r"""
Return a representative for each conjugacy class for the
discriminant ``D`` (ignoring the sign).
If ``primitive=True`` only one representative for each
fixed point is returned (ignoring sign).
INPUT:
- ``D`` -- An element of the base ring corresponding
to a valid discriminant.
- ``primitive`` -- If ``True`` (default) then only primitive
representatives are considered.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=4)
sage: G.element_repr_method("conj")
sage: R = G.class_representatives(4)
sage: R
[[V(2)]]
sage: [v.continued_fraction()[1] for v in R]
[(2,)]
sage: R = G.class_representatives(0)
sage: R
[[V(3)]]
sage: [v.continued_fraction()[1] for v in R]
[(1, 2)]
sage: R = G.class_representatives(-4)
sage: R
[[S]]
sage: R = G.class_representatives(-4, primitive=False)
sage: R
[[S], [U^2]]
sage: R = G.class_representatives(G.lam()^2 - 4)
sage: R
[[U]]
sage: R = G.class_representatives(G.lam()^2 - 4, primitive=False)
sage: R
[[U], [U^(-1)]]
sage: R = G.class_representatives(14)
sage: R
[[V(2)*V(3)], [V(1)*V(2)]]
sage: [v.continued_fraction()[1] for v in R]
[(1, 2, 2), (3,)]
sage: R = G.class_representatives(32)
sage: R
[[V(3)^2*V(1)], [V(1)^2*V(3)]]
sage: [v.continued_fraction()[1] for v in R]
[(1, 2, 1, 3), (1, 4)]
sage: R = G.class_representatives(32, primitive=False)
sage: R
[[V(3)^2*V(1)], [V(1)^2*V(3)], [V(2)^2]]
sage: G.element_repr_method("default")
"""
if coerce_AA(D) == 0 and not primitive:
raise ValueError(
"There are infinitely many non-primitive conjugacy classes of discriminant 0."
)
self._conjugacy_representatives(D=D)
L = []
if D in self._conj_prim:
L += self._conj_prim[D]
if not primitive and D in self._conj_nonprim:
L += self._conj_nonprim[D]
if len(L) == 0:
raise ValueError("D = {} is not a{} discriminant for {}".format(
D, " primitive" if primitive else "", self))
else:
return L
def _conjugacy_representatives(self, max_block_length=ZZ(0), D=None):
r"""
Store conjugacy representatives up to block length
``max_block_length`` (a non-negative integer, default: 0)
in the internal dictionary. Previously calculated data is reused.
This is a helper function for e.g. :meth:`class_number`.
The set of all (hyperbolic) conjugacy types of block length
``t`` is stored in ``self._conj_block[t]``.
The set of all primitive representatives (so far) with
discriminant ``D`` is stored in ``self._conj_prim[D]``.
The set of all non-primitive representatives (so far) with
discriminant ``D`` is stored in ``self._conj_nonprim[D]``.
The case of non-positive discriminants is done manually.
INPUT:
- ``max_block_length`` -- A non-negative integer (default: ``0``),
the maximal block length.
- ``D`` -- An element/discriminant of the base ring or
more generally an upper bound for the
involved discriminants. If ``D != None``
then an upper bound for ``max_block_length``
is deduced from ``D`` (default: ``None``).
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=5)
sage: G.element_repr_method("conj")
sage: G._conjugacy_representatives(2)
sage: list(G._conj_block[2])
[((4, 1), (3, 1)), ((2, 2),), ((3, 2),), ((3, 1), (1, 1)), ((4, 1), (1, 1)), ((4, 1), (2, 1)), ((3, 1), (2, 1)), ((2, 1), (1, 1))]
sage: [key for key in G._conj_prim]
[0, lam - 3, 15*lam + 6, 7*lam + 6, 4*lam, 9*lam + 5, 33*lam + 21, -4]
sage: for key in G._conj_prim: print G._conj_prim[key]
[[V(4)]]
[[U], [U]]
[[V(1)*V(3)], [V(2)*V(4)]]
[[V(1)*V(4)]]
[[V(3)], [V(2)]]
[[V(3)*V(4)], [V(1)*V(2)]]
[[V(2)*V(3)]]
[[S], [S]]
sage: [key for key in G._conj_nonprim]
[lam - 3, 32*lam + 16, -lam - 2]
sage: for key in G._conj_nonprim: print G._conj_nonprim[key]
[[U^(-1)], [U^(-1)]]
[[V(2)^2], [V(3)^2]]
[[U^(-2)], [U^2], [U^(-2)], [U^2]]
sage: G.element_repr_method("default")
"""
from sage.combinat.partition import OrderedPartitions
from sage.combinat.combinat import tuples
from sage.rings.arith import divisors
if not D is None:
max_block_length = max(coerce_AA(0),
coerce_AA((D + 4) /
(self.lam()**2))).sqrt().floor()
else:
try:
max_block_length = ZZ(max_block_length)
if max_block_length < 0:
raise TypeError
except TypeError:
raise ValueError(
"max_block_length must be a non-negative integer!")
if not hasattr(self, "_max_block_length"):
self._max_block_length = ZZ(0)
self._conj_block = {}
self._conj_nonprim = {}
self._conj_prim = {}
# It is not clear how to define the class number for D=0:
# Conjugacy classes are V(n-1)^(+-k) for arbitrary k
# and the trivial class (what about self_conj_block[0]?).
#
# One way is to define it using the fixed points and in
# that case V(n-1) would be a good representative.
# The non-primitive case is unclear however...
#
# We set it here to ensure that 0 is enlisted as a discriminant...
#
self._conj_prim[ZZ(0)] = []
self._conj_prim[ZZ(0)].append(self.V(self.n() - 1))
self._elliptic_conj_reps()
if max_block_length <= self._max_block_length:
return
def is_cycle(seq):
length = len(seq)
for n in divisors(length):
if n < length and is_cycle_of_length(seq, n):
return True
return False
def is_cycle_of_length(seq, n):
for i in range(n, len(seq)):
if seq[i] != seq[i % n]:
return False
return True
j_list = range(1, self.n())
for t in range(self._max_block_length + 1, max_block_length + 1):
t_ZZ = ZZ(t)
if t_ZZ not in self._conj_block:
self._conj_block[t_ZZ] = set()
partitions = OrderedPartitions(t).list()
for par in partitions:
len_par = len(par)
exp_list = tuples(j_list, len_par)
for ex in exp_list:
keep = True
if len_par > 1:
for k in range(-1, len_par - 1):
if ex[k] == ex[k + 1]:
keep = False
break
# We don't want powers of V(1)
elif ex[0] == 1:
keep = False
# But: Do we maybe want powers of V(n-1)??
# For now we exclude the parabolic cases...
elif ex[0] == self.n() - 1:
keep = False
if keep:
conj_type = cyclic_representative(
tuple((ZZ(ex[k]), ZZ(par[k]))
for k in range(len_par)))
self._conj_block[t_ZZ].add(conj_type)
for el in self._conj_block[t_ZZ]:
group_el = prod(
[self.V(el[k][0])**el[k][1] for k in range(len(el))])
#if el != group_el.conjugacy_type():
# raise AssertionError("This shouldn't happen!")
D = group_el.discriminant()
if coerce_AA(D) < 0:
raise AssertionError("This shouldn't happen!")
if coerce_AA(D) == 0:
raise AssertionError("This shouldn't happen!")
#continue
# The primitive cases
#if group_el.is_primitive():
if not ((len(el) == 1 and el[0][1] > 1) or is_cycle(el)):
if D not in self._conj_prim:
self._conj_prim[D] = []
self._conj_prim[D].append(group_el)
# The remaining cases
else:
if D not in self._conj_nonprim:
self._conj_nonprim[D] = []
self._conj_nonprim[D].append(group_el)
self._max_block_length = max_block_length
def class_representatives(self, D, primitive=True):
r"""
Return a representative for each conjugacy class for the
discriminant ``D`` (ignoring the sign).
If ``primitive=True`` only one representative for each
fixed point is returned (ignoring sign).
INPUT:
- ``D`` -- An element of the base ring corresponding
to a valid discriminant.
- ``primitive`` -- If ``True`` (default) then only primitive
representatives are considered.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=4)
sage: G.element_repr_method("conj")
sage: R = G.class_representatives(4)
sage: R
[[V(2)]]
sage: [v.continued_fraction()[1] for v in R]
[(2,)]
sage: R = G.class_representatives(0)
sage: R
[[V(3)]]
sage: [v.continued_fraction()[1] for v in R]
[(1, 2)]
sage: R = G.class_representatives(-4)
sage: R
[[S]]
sage: R = G.class_representatives(-4, primitive=False)
sage: R
[[S], [U^2]]
sage: R = G.class_representatives(G.lam()^2 - 4)
sage: R
[[U]]
sage: R = G.class_representatives(G.lam()^2 - 4, primitive=False)
sage: R
[[U], [U^(-1)]]
sage: R = G.class_representatives(14)
sage: R
[[V(2)*V(3)], [V(1)*V(2)]]
sage: [v.continued_fraction()[1] for v in R]
[(1, 2, 2), (3,)]
sage: R = G.class_representatives(32)
sage: R
[[V(3)^2*V(1)], [V(1)^2*V(3)]]
sage: [v.continued_fraction()[1] for v in R]
[(1, 2, 1, 3), (1, 4)]
sage: R = G.class_representatives(32, primitive=False)
sage: R
[[V(3)^2*V(1)], [V(1)^2*V(3)], [V(2)^2]]
sage: G.element_repr_method("default")
"""
if coerce_AA(D) == 0 and not primitive:
raise ValueError("There are infinitely many non-primitive conjugacy classes of discriminant 0.")
self._conjugacy_representatives(D=D)
L = []
if D in self._conj_prim:
L += self._conj_prim[D]
if not primitive and D in self._conj_nonprim:
L += self._conj_nonprim[D]
if len(L) == 0:
raise ValueError("D = {} is not a{} discriminant for {}".format(D, " primitive" if primitive else "", self))
else:
return L