def simplicial_complex(self):
r"""
Return a simplicial complex from a triangulation of the point
configuration.
OUTPUT:
A :class:`~sage.homology.simplicial_complex.SimplicialComplex`.
EXAMPLES::
sage: p = polytopes.cuboctahedron()
sage: sc = p.triangulate(engine='internal').simplicial_complex()
sage: sc
Simplicial complex with 12 vertices and 16 facets
Any convex set is contractable, so its reduced homology groups vanish::
sage: sc.homology()
{0: 0, 1: 0, 2: 0, 3: 0}
"""
from sage.homology.simplicial_complex import SimplicialComplex
from sage.misc.all import flatten
vertex_set = set(flatten(self))
return SimplicialComplex(vertex_set = vertex_set,
maximal_faces = self)
def _split_hyperbolic(L) :
cur_cor = 2
Lcor = L + cur_cor * identity_matrix(L.nrows())
while not is_positive_definite(Lcor) :
cur_cor += 2
Lcor = L + cur_cor * identity_matrix(L.nrows())
a = FreeModule(ZZ, L.nrows()).gen(0)
if a * L * a >= 0 :
Lcor_length_inc = max(3, a * L * a)
cur_Lcor_length = Lcor_length_inc
while True :
short_vectors = flatten(QuadraticForm(Lcor).short_vector_list_up_to_length( a * Lcor * a )[cur_Lcor_length - Lcor_length_inc: cur_Lcor_length], max_level = 1)
for a in short_vectors :
if a * L * a < 0 :
break
else :
continue
break
n = -a * L * a // 2
short_vectors = E8.short_vector_list_up_to_length(n + 1)[-1]
for v in short_vectors :
for w in short_vectors :
if v * E8_gram * w == 2 * n - 1 :
LE8_mat = L.block_sum(E8_gram)
v_form = vector( list(a) + list(v) ) * LE8_mat
w_form = vector( list(a) + list(w) ) * LE8_mat
Lred_basis = matrix(ZZ, [v_form, w_form]).right_kernel().basis_matrix().transpose()
Lred_basis = matrix(ZZ, Lred_basis)
return Lred_basis.transpose() * LE8_mat * Lred_basis
def __init__(self, toric_variety, base_ring, check):
r"""
EXAMPLES::
sage: from sage.schemes.toric.chow_group import *
sage: P2=toric_varieties.P2()
sage: A = ChowGroup_class(P2,ZZ,True); A
Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches
sage: is_ChowGroup(A)
True
sage: is_ChowCycle(A.an_element())
True
TESTS::
sage: A_ZZ = P2.Chow_group()
sage: 2 * A_ZZ.an_element() * 3
( 6 | 0 | 0 )
sage: 1/2 * A_ZZ.an_element() * 1/3
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '*': 'Rational Field'
and 'Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches'
sage: A_ZZ.get_action(ZZ)
Right scalar multiplication by Integer Ring on Chow group of 2-d
CPR-Fano toric variety covered by 3 affine patches
sage: A_ZZ.get_action(QQ)
You can't multiply integer classes with fractional
numbers. For that you need to go to the rational Chow group::
sage: A_QQ = P2.Chow_group(QQ)
sage: 2 * A_QQ.an_element() * 3
( 0 | 0 | 6 )
sage: 1/2 * A_QQ.an_element() * 1/3
( 0 | 0 | 1/6 )
sage: A_QQ.get_action(ZZ)
Right scalar multiplication by Integer Ring on QQ-Chow group of 2-d
CPR-Fano toric variety covered by 3 affine patches
sage: A_QQ.get_action(QQ)
Right scalar multiplication by Rational Field on QQ-Chow group of 2-d
CPR-Fano toric variety covered by 3 affine patches
"""
self._variety = toric_variety
# cones are automatically sorted by dimension
self._cones = flatten( toric_variety.fan().cones() )
V = FreeModule(base_ring, len(self._cones))
W = self._rational_equivalence_relations(V)
super(ChowGroup_class,self).__init__(V, W, check)
def __init__(self, toric_variety, base_ring, check):
r"""
EXAMPLES::
sage: from sage.schemes.toric.chow_group import *
sage: P2=toric_varieties.P2()
sage: A = ChowGroup_class(P2,ZZ,True); A
Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches
sage: is_ChowGroup(A)
True
sage: is_ChowCycle(A.an_element())
True
TESTS::
sage: A_ZZ = P2.Chow_group()
sage: 2 * A_ZZ.an_element() * 3
( 6 | 0 | 0 )
sage: 1/2 * A_ZZ.an_element() * 1/3
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: 'Rational Field'
and 'Chow group of 2-d CPR-Fano toric variety covered by 3 affine patches'
sage: A_ZZ.get_action(ZZ)
Right scalar multiplication by Integer Ring on Chow group of 2-d
CPR-Fano toric variety covered by 3 affine patches
sage: A_ZZ.get_action(QQ)
You can't multiply integer classes with fractional
numbers. For that you need to go to the rational Chow group::
sage: A_QQ = P2.Chow_group(QQ)
sage: 2 * A_QQ.an_element() * 3
( 0 | 0 | 6 )
sage: 1/2 * A_QQ.an_element() * 1/3
( 0 | 0 | 1/6 )
sage: A_QQ.get_action(ZZ)
Right scalar multiplication by Integer Ring on QQ-Chow group of 2-d
CPR-Fano toric variety covered by 3 affine patches
sage: A_QQ.get_action(QQ)
Right scalar multiplication by Rational Field on QQ-Chow group of 2-d
CPR-Fano toric variety covered by 3 affine patches
"""
self._variety = toric_variety
# cones are automatically sorted by dimension
self._cones = flatten( toric_variety.fan().cones() )
V = FreeModule(base_ring, len(self._cones))
W = self._rational_equivalence_relations(V)
super(ChowGroup_class,self).__init__(V, W, check)
def _split_hyperbolic(L):
cur_cor = 2
Lcor = L + cur_cor * identity_matrix(L.nrows())
while not is_positive_definite(Lcor):
cur_cor += 2
Lcor = L + cur_cor * identity_matrix(L.nrows())
a = FreeModule(ZZ, L.nrows()).gen(0)
if a * L * a >= 0:
Lcor_length_inc = max(3, a * L * a)
cur_Lcor_length = Lcor_length_inc
while True:
short_vectors = flatten(
QuadraticForm(Lcor).short_vector_list_up_to_length(
a * Lcor * a)[cur_Lcor_length -
Lcor_length_inc:cur_Lcor_length],
max_level=1)
for a in short_vectors:
if a * L * a < 0:
break
else:
continue
break
n = -a * L * a // 2
short_vectors = E8.short_vector_list_up_to_length(n + 1)[-1]
for v in short_vectors:
for w in short_vectors:
if v * E8_gram * w == 2 * n - 1:
LE8_mat = L.block_sum(E8_gram)
v_form = vector(list(a) + list(v)) * LE8_mat
w_form = vector(list(a) + list(w)) * LE8_mat
Lred_basis = matrix(
ZZ, [v_form, w_form
]).right_kernel().basis_matrix().transpose()
Lred_basis = matrix(ZZ, Lred_basis)
return Lred_basis.transpose() * LE8_mat * Lred_basis
def _compute(self, verbose=False):
"""
Computes the list of curves, the matrix and prime-degree
isogenies.
EXAMPLES::
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve(K, [0,0,0,0,1])
sage: C = E.isogeny_class()
sage: C2 = C.copy()
sage: C2._mat
sage: C2._compute()
sage: C2._mat
[1 3 6 2]
[3 1 2 6]
[6 2 1 3]
[2 6 3 1]
sage: C2._compute(verbose=True)
possible isogeny degrees: [2, 3] -actual isogeny degrees: {2, 3} -added curve #1 (degree 2)... -added tuple [0, 1, 2]... -added tuple [1, 0, 2]... -added curve #2 (degree 3)... -added tuple [0, 2, 3]... -added tuple [2, 0, 3]...... relevant degrees: [2, 3]... -now completing the isogeny class... -processing curve #1... -added tuple [1, 0, 2]... -added tuple [0, 1, 2]... -added curve #3... -added tuple [1, 3, 3]... -added tuple [3, 1, 3]... -processing curve #2... -added tuple [2, 3, 2]... -added tuple [3, 2, 2]... -added tuple [2, 0, 3]... -added tuple [0, 2, 3]... -processing curve #3... -added tuple [3, 2, 2]... -added tuple [2, 3, 2]... -added tuple [3, 1, 3]... -added tuple [1, 3, 3]...... isogeny class has size 4
Sorting permutation = {0: 1, 1: 2, 2: 0, 3: 3}
Matrix = [1 3 6 2]
[3 1 2 6]
[6 2 1 3]
[2 6 3 1]
TESTS:
Check that :trac:`19030` is fixed (codomains of reverse isogenies were wrong)::
sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve([1, i + 1, 1, -72*i + 8, 95*i + 146])
sage: C = E.isogeny_class()
sage: curves = C.curves
sage: isos = C.isogenies()
sage: isos[0][3].codomain() == curves[3]
True
"""
from sage.schemes.elliptic_curves.ell_curve_isogeny import fill_isogeny_matrix, unfill_isogeny_matrix
from sage.matrix.all import MatrixSpace
from sage.sets.set import Set
self._maps = None
E = self.E.global_minimal_model(semi_global=True)
degs = possible_isogeny_degrees(E)
if verbose:
import sys
sys.stdout.write(" possible isogeny degrees: %s" % degs)
sys.stdout.flush()
isogenies = E.isogenies_prime_degree(degs)
if verbose:
sys.stdout.write(" -actual isogeny degrees: %s" % Set([phi.degree() for phi in isogenies]))
sys.stdout.flush()
# Add all new codomains to the list and collect degrees:
curves = [E]
ncurves = 1
degs = []
# tuples (i,j,l,phi) where curve i is l-isogenous to curve j via phi
tuples = []
def add_tup(t):
for T in [t, [t[1],t[0],t[2],0]]:
if not T in tuples:
tuples.append(T)
if verbose:
sys.stdout.write(" -added tuple %s..." % T[:3])
sys.stdout.flush()
for phi in isogenies:
E2 = phi.codomain()
d = ZZ(phi.degree())
if not any([E2.is_isomorphic(E3) for E3 in curves]):
curves.append(E2)
if verbose:
sys.stdout.write(" -added curve #%s (degree %s)..." % (ncurves,d))
sys.stdout.flush()
add_tup([0,ncurves,d,phi])
ncurves += 1
if not d in degs:
degs.append(d)
if verbose:
sys.stdout.write("... relevant degrees: %s..." % degs)
sys.stdout.write(" -now completing the isogeny class...")
sys.stdout.flush()
i = 1
while i < ncurves:
E1 = curves[i]
if verbose:
sys.stdout.write(" -processing curve #%s..." % i)
sys.stdout.flush()
isogenies = E1.isogenies_prime_degree(degs)
for phi in isogenies:
E2 = phi.codomain()
d = phi.degree()
js = [j for j,E3 in enumerate(curves) if E2.is_isomorphic(E3)]
if js: # seen codomain already -- up to isomorphism
j = js[0]
if phi.codomain()!=curves[j]:
iso = E2.isomorphism_to(curves[j])
phi.set_post_isomorphism(iso)
assert phi.domain()==curves[i] and phi.codomain()==curves[j]
add_tup([i,j,d,phi])
else:
curves.append(E2)
if verbose:
sys.stdout.write(" -added curve #%s..." % ncurves)
sys.stdout.flush()
add_tup([i,ncurves,d,phi])
ncurves += 1
i += 1
if verbose:
print("... isogeny class has size %s" % ncurves)
# key function for sorting
if E.has_rational_cm():
key_function = lambda E: (-E.cm_discriminant(),flatten([list(ai) for ai in E.ainvs()]))
else:
key_function = lambda E: flatten([list(ai) for ai in E.ainvs()])
self.curves = sorted(curves,key=key_function)
perm = dict([(i,self.curves.index(E)) for i,E in enumerate(curves)])
if verbose:
print "Sorting permutation = %s" % perm
mat = MatrixSpace(ZZ,ncurves)(0)
self._maps = [[0]*ncurves for i in range(ncurves)]
for i,j,l,phi in tuples:
if phi!=0:
mat[perm[i],perm[j]] = l
self._maps[perm[i]][perm[j]] = phi
self._mat = fill_isogeny_matrix(mat)
if verbose:
print "Matrix = %s" % self._mat
if not E.has_rational_cm():
self._qfmat = None
return
# In the CM case, we will have found some "horizontal"
# isogenies of composite degree and would like to replace them
# by isogenies of prime degree, mainly to make the isogeny
# graph look better. We also construct a matrix whose entries
# are not degrees of cyclic isogenies, but rather quadratic
# forms (in 1 or 2 variables) representing the isogeny
# degrees. For this we take a short cut: properly speaking,
# when `\text{End}(E_1)=\text{End}(E_2)=O`, the set
# `\text{Hom}(E_1,E_2)` is a rank `1` projective `O`-module,
# hence has a well-defined ideal class associated to it, and
# hence (using an identification between the ideal class group
# and the group of classes of primitive quadratic forms of the
# same discriminant) an equivalence class of quadratic forms.
# But we currently only care about the numbers represented by
# the form, i.e. which genus it is in rather than the exact
# class. So it suffices to find one form of the correct
# discriminant which represents one isogeny degree from `E_1`
# to `E_2` in order to obtain a form which represents all such
# degrees.
if verbose:
print("Creating degree matrix (CM case)")
allQs = {} # keys: discriminants d
# values: lists of equivalence classes of
# primitive forms of discriminant d
def find_quadratic_form(d,n):
if not d in allQs:
from sage.quadratic_forms.binary_qf import BinaryQF_reduced_representatives
allQs[d] = BinaryQF_reduced_representatives(d, primitive_only=True)
# now test which of the Qs represents n
for Q in allQs[d]:
if Q.solve_integer(n):
return Q
raise ValueError("No form of discriminant %d represents %s" %(d,n))
mat = self._mat
qfmat = [[0 for i in range(ncurves)] for j in range(ncurves)]
for i, E1 in enumerate(self.curves):
for j, E2 in enumerate(self.curves):
if j<i:
qfmat[i][j] = qfmat[j][i]
mat[i,j] = mat[j,i]
elif i==j:
qfmat[i][j] = [1]
# mat[i,j] already 1
else:
d = E1.cm_discriminant()
if d != E2.cm_discriminant():
qfmat[i][j] = [mat[i,j]]
# mat[i,j] already unique
else: # horizontal isogeny
q = find_quadratic_form(d,mat[i,j])
qfmat[i][j] = list(q)
mat[i,j] = q.small_prime_value()
self._mat = mat
self._qfmat = qfmat
if verbose:
print("new matrix = %s" % mat)
print("matrix of forms = %s" % qfmat)
def _compute(self, verbose=False):
"""
Computes the list of curves, the matrix and prime-degree
isogenies.
EXAMPLES::
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve(K, [0,0,0,0,1])
sage: C = E.isogeny_class()
sage: C2 = C.copy()
sage: C2._mat
sage: C2._compute()
sage: C2._mat
[1 3 6 2]
[3 1 2 6]
[6 2 1 3]
[2 6 3 1]
sage: C2._compute(verbose=True)
possible isogeny degrees: [2, 3] -actual isogeny degrees: {2, 3} -added curve #1 (degree 2)... -added tuple [0, 1, 2]... -added tuple [1, 0, 2]... -added curve #2 (degree 3)... -added tuple [0, 2, 3]... -added tuple [2, 0, 3]...... relevant degrees: [2, 3]... -now completing the isogeny class... -processing curve #1... -added tuple [1, 0, 2]... -added tuple [0, 1, 2]... -added curve #3... -added tuple [1, 3, 3]... -added tuple [3, 1, 3]... -processing curve #2... -added tuple [2, 3, 2]... -added tuple [3, 2, 2]... -added tuple [2, 0, 3]... -added tuple [0, 2, 3]... -processing curve #3... -added tuple [3, 2, 2]... -added tuple [2, 3, 2]... -added tuple [3, 1, 3]... -added tuple [1, 3, 3]...... isogeny class has size 4
Sorting permutation = {0: 1, 1: 2, 2: 0, 3: 3}
Matrix = [1 3 6 2]
[3 1 2 6]
[6 2 1 3]
[2 6 3 1]
TESTS:
Check that :trac:`19030` is fixed (codomains of reverse isogenies were wrong)::
sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve([1, i + 1, 1, -72*i + 8, 95*i + 146])
sage: C = E.isogeny_class()
sage: curves = C.curves
sage: isos = C.isogenies()
sage: isos[0][3].codomain() == curves[3]
True
"""
from sage.schemes.elliptic_curves.ell_curve_isogeny import fill_isogeny_matrix, unfill_isogeny_matrix
from sage.matrix.all import MatrixSpace
from sage.sets.set import Set
self._maps = None
E = self.E.global_minimal_model(semi_global=True)
degs = possible_isogeny_degrees(E)
if verbose:
import sys
sys.stdout.write(" possible isogeny degrees: %s" % degs)
sys.stdout.flush()
isogenies = E.isogenies_prime_degree(degs)
if verbose:
sys.stdout.write(" -actual isogeny degrees: %s" % Set([phi.degree() for phi in isogenies]))
sys.stdout.flush()
# Add all new codomains to the list and collect degrees:
curves = [E]
ncurves = 1
degs = []
# tuples (i,j,l,phi) where curve i is l-isogenous to curve j via phi
tuples = []
def add_tup(t):
for T in [t, [t[1],t[0],t[2],0]]:
if not T in tuples:
tuples.append(T)
if verbose:
sys.stdout.write(" -added tuple %s..." % T[:3])
sys.stdout.flush()
for phi in isogenies:
E2 = phi.codomain()
d = ZZ(phi.degree())
if not any([E2.is_isomorphic(E3) for E3 in curves]):
curves.append(E2)
if verbose:
sys.stdout.write(" -added curve #%s (degree %s)..." % (ncurves,d))
sys.stdout.flush()
add_tup([0,ncurves,d,phi])
ncurves += 1
if not d in degs:
degs.append(d)
if verbose:
sys.stdout.write("... relevant degrees: %s..." % degs)
sys.stdout.write(" -now completing the isogeny class...")
sys.stdout.flush()
i = 1
while i < ncurves:
E1 = curves[i]
if verbose:
sys.stdout.write(" -processing curve #%s..." % i)
sys.stdout.flush()
isogenies = E1.isogenies_prime_degree(degs)
for phi in isogenies:
E2 = phi.codomain()
d = phi.degree()
js = [j for j,E3 in enumerate(curves) if E2.is_isomorphic(E3)]
if js: # seen codomain already -- up to isomorphism
j = js[0]
if phi.codomain()!=curves[j]:
iso = E2.isomorphism_to(curves[j])
phi.set_post_isomorphism(iso)
assert phi.domain()==curves[i] and phi.codomain()==curves[j]
add_tup([i,j,d,phi])
else:
curves.append(E2)
if verbose:
sys.stdout.write(" -added curve #%s..." % ncurves)
sys.stdout.flush()
add_tup([i,ncurves,d,phi])
ncurves += 1
i += 1
if verbose:
print("... isogeny class has size %s" % ncurves)
# key function for sorting
if E.has_rational_cm():
key_function = lambda E: (-E.cm_discriminant(),flatten([list(ai) for ai in E.ainvs()]))
else:
key_function = lambda E: flatten([list(ai) for ai in E.ainvs()])
self.curves = sorted(curves,key=key_function)
perm = dict([(i,self.curves.index(E)) for i,E in enumerate(curves)])
if verbose:
print("Sorting permutation = %s" % perm)
mat = MatrixSpace(ZZ,ncurves)(0)
self._maps = [[0]*ncurves for i in range(ncurves)]
for i,j,l,phi in tuples:
if phi!=0:
mat[perm[i],perm[j]] = l
self._maps[perm[i]][perm[j]] = phi
self._mat = fill_isogeny_matrix(mat)
if verbose:
print("Matrix = %s" % self._mat)
if not E.has_rational_cm():
self._qfmat = None
return
# In the CM case, we will have found some "horizontal"
# isogenies of composite degree and would like to replace them
# by isogenies of prime degree, mainly to make the isogeny
# graph look better. We also construct a matrix whose entries
# are not degrees of cyclic isogenies, but rather quadratic
# forms (in 1 or 2 variables) representing the isogeny
# degrees. For this we take a short cut: properly speaking,
# when `\text{End}(E_1)=\text{End}(E_2)=O`, the set
# `\text{Hom}(E_1,E_2)` is a rank `1` projective `O`-module,
# hence has a well-defined ideal class associated to it, and
# hence (using an identification between the ideal class group
# and the group of classes of primitive quadratic forms of the
# same discriminant) an equivalence class of quadratic forms.
# But we currently only care about the numbers represented by
# the form, i.e. which genus it is in rather than the exact
# class. So it suffices to find one form of the correct
# discriminant which represents one isogeny degree from `E_1`
# to `E_2` in order to obtain a form which represents all such
# degrees.
if verbose:
print("Creating degree matrix (CM case)")
allQs = {} # keys: discriminants d
# values: lists of equivalence classes of
# primitive forms of discriminant d
def find_quadratic_form(d,n):
if not d in allQs:
from sage.quadratic_forms.binary_qf import BinaryQF_reduced_representatives
allQs[d] = BinaryQF_reduced_representatives(d, primitive_only=True)
# now test which of the Qs represents n
for Q in allQs[d]:
if Q.solve_integer(n):
return Q
raise ValueError("No form of discriminant %d represents %s" %(d,n))
mat = self._mat
qfmat = [[0 for i in range(ncurves)] for j in range(ncurves)]
for i, E1 in enumerate(self.curves):
for j, E2 in enumerate(self.curves):
if j<i:
qfmat[i][j] = qfmat[j][i]
mat[i,j] = mat[j,i]
elif i==j:
qfmat[i][j] = [1]
# mat[i,j] already 1
else:
d = E1.cm_discriminant()
if d != E2.cm_discriminant():
qfmat[i][j] = [mat[i,j]]
# mat[i,j] already unique
else: # horizontal isogeny
q = find_quadratic_form(d,mat[i,j])
qfmat[i][j] = list(q)
mat[i,j] = q.small_prime_value()
self._mat = mat
self._qfmat = qfmat
if verbose:
print("new matrix = %s" % mat)
print("matrix of forms = %s" % qfmat)