def matrix(self, fill=True):
"""
Returns the matrix whose entries give the minimal degrees of
isogenies between curves in this class.
INPUT:
- ``fill`` -- boolean (default True). If False then the
matrix will contain only zeros and prime entries; if True it
will fill in the other degrees.
EXAMPLES::
sage: isocls = EllipticCurve('15a3').isogeny_class(use_tuple=False)
sage: isocls.matrix()
[ 1 2 2 2 4 4 8 8]
[ 2 1 4 4 8 8 16 16]
[ 2 4 1 4 8 8 16 16]
[ 2 4 4 1 2 2 4 4]
[ 4 8 8 2 1 4 8 8]
[ 4 8 8 2 4 1 2 2]
[ 8 16 16 4 8 2 1 4]
[ 8 16 16 4 8 2 4 1]
sage: isocls.matrix(fill=False)
[0 2 2 2 0 0 0 0]
[2 0 0 0 0 0 0 0]
[2 0 0 0 0 0 0 0]
[2 0 0 0 2 2 0 0]
[0 0 0 2 0 0 0 0]
[0 0 0 2 0 0 2 2]
[0 0 0 0 0 2 0 0]
[0 0 0 0 0 2 0 0]
"""
if self._mat is None:
self._compute_matrix()
mat = self._mat
if fill and mat[0,0] == 0:
from sage.schemes.elliptic_curves.ell_curve_isogeny import fill_isogeny_matrix
mat = fill_isogeny_matrix(mat)
if not fill and mat[0,0] == 1:
from sage.schemes.elliptic_curves.ell_curve_isogeny import unfill_isogeny_matrix
mat = unfill_isogeny_matrix(mat)
return mat
def matrix(self, fill=True):
"""
Returns the matrix whose entries give the minimal degrees of
isogenies between curves in this class.
INPUT:
- ``fill`` -- boolean (default True). If False then the
matrix will contain only zeros and prime entries; if True it
will fill in the other degrees.
EXAMPLES::
sage: isocls = EllipticCurve('15a3').isogeny_class(use_tuple=False)
sage: isocls.matrix()
[ 1 2 2 2 4 4 8 8]
[ 2 1 4 4 8 8 16 16]
[ 2 4 1 4 8 8 16 16]
[ 2 4 4 1 2 2 4 4]
[ 4 8 8 2 1 4 8 8]
[ 4 8 8 2 4 1 2 2]
[ 8 16 16 4 8 2 1 4]
[ 8 16 16 4 8 2 4 1]
sage: isocls.matrix(fill=False)
[0 2 2 2 0 0 0 0]
[2 0 0 0 0 0 0 0]
[2 0 0 0 0 0 0 0]
[2 0 0 0 2 2 0 0]
[0 0 0 2 0 0 0 0]
[0 0 0 2 0 0 2 2]
[0 0 0 0 0 2 0 0]
[0 0 0 0 0 2 0 0]
"""
if self._mat is None:
self._compute_matrix()
mat = self._mat
if fill and mat[0, 0] == 0:
from sage.schemes.elliptic_curves.ell_curve_isogeny import fill_isogeny_matrix
mat = fill_isogeny_matrix(mat)
if not fill and mat[0, 0] == 1:
from sage.schemes.elliptic_curves.ell_curve_isogeny import unfill_isogeny_matrix
mat = unfill_isogeny_matrix(mat)
return mat
def make_graph(M):
"""
Code extracted from Sage's elliptic curve isogeny class (reshaped
in the case maxdegree==12)
"""
from sage.schemes.elliptic_curves.ell_curve_isogeny import fill_isogeny_matrix, unfill_isogeny_matrix
from sage.graphs.graph import Graph
n = M.nrows() # = M.ncols()
G = Graph(unfill_isogeny_matrix(M), format='weighted_adjacency_matrix')
MM = fill_isogeny_matrix(M)
# The maximum degree classifies the shape of the isogeny
# graph, though the number of vertices is often enough.
# This only holds over Q, so this code will need to change
# once other isogeny classes are implemented.
if n == 1:
# one vertex
pass
elif n == 2:
# one edge, two vertices. We align horizontally and put
# the lower number on the left vertex.
G.set_pos(pos={0:[-0.5,0],1:[0.5,0]})
else:
maxdegree = max(max(MM))
if n == 3:
# o--o--o
centervert = [i for i in range(3) if max(MM.row(i)) < maxdegree][0]
other = [i for i in range(3) if i != centervert]
G.set_pos(pos={centervert:[0,0],other[0]:[-1,0],other[1]:[1,0]})
elif maxdegree == 4:
# o--o<8
centervert = [i for i in range(4) if max(MM.row(i)) < maxdegree][0]
other = [i for i in range(4) if i != centervert]
G.set_pos(pos={centervert:[0,0],other[0]:[0,1],other[1]:[-0.8660254,-0.5],other[2]:[0.8660254,-0.5]})
elif maxdegree == 27:
# o--o--o--o
centers = [i for i in range(4) if list(MM.row(i)).count(3) == 2]
left = [j for j in range(4) if MM[centers[0],j] == 3 and j not in centers][0]
right = [j for j in range(4) if MM[centers[1],j] == 3 and j not in centers][0]
G.set_pos(pos={left:[-1.5,0],centers[0]:[-0.5,0],centers[1]:[0.5,0],right:[1.5,0]})
elif n == 4:
# square
opp = [i for i in range(1,4) if not MM[0,i].is_prime()][0]
other = [i for i in range(1,4) if i != opp]
G.set_pos(pos={0:[1,1],other[0]:[-1,1],opp:[-1,-1],other[1]:[1,-1]})
elif maxdegree == 8:
# 8>o--o<8
centers = [i for i in range(6) if list(MM.row(i)).count(2) == 3]
left = [j for j in range(6) if MM[centers[0],j] == 2 and j not in centers]
right = [j for j in range(6) if MM[centers[1],j] == 2 and j not in centers]
G.set_pos(pos={centers[0]:[-0.5,0],left[0]:[-1,0.8660254],left[1]:[-1,-0.8660254],centers[1]:[0.5,0],right[0]:[1,0.8660254],right[1]:[1,-0.8660254]})
elif maxdegree == 18:
# two squares joined on an edge
centers = [i for i in range(6) if list(MM.row(i)).count(3) == 2]
top = [j for j in range(6) if MM[centers[0],j] == 3]
bl = [j for j in range(6) if MM[top[0],j] == 2][0]
br = [j for j in range(6) if MM[top[1],j] == 2][0]
G.set_pos(pos={centers[0]:[0,0.5],centers[1]:[0,-0.5],top[0]:[-1,0.5],top[1]:[1,0.5],bl:[-1,-0.5],br:[1,-0.5]})
elif maxdegree == 16:
# tree from bottom, 3 regular except for the leaves.
centers = [i for i in range(8) if list(MM.row(i)).count(2) == 3]
center = [i for i in centers if len([j for j in centers if MM[i,j] == 2]) == 2][0]
centers.remove(center)
bottom = [j for j in range(8) if MM[center,j] == 2 and j not in centers][0]
left = [j for j in range(8) if MM[centers[0],j] == 2 and j != center]
right = [j for j in range(8) if MM[centers[1],j] == 2 and j != center]
G.set_pos(pos={center:[0,0],bottom:[0,-1],centers[0]:[-0.8660254,0.5],centers[1]:[0.8660254,0.5],left[0]:[-0.8660254,1.5],right[0]:[0.8660254,1.5],left[1]:[-1.7320508,0],right[1]:[1.7320508,0]})
elif maxdegree == 12:
# tent
centers = [i for i in range(8) if list(MM.row(i)).count(2) == 3]
left = [j for j in range(8) if MM[centers[0],j] == 2]
right = []
for i in range(3):
right.append([j for j in range(8) if MM[centers[1],j] == 2 and MM[left[i],j] == 3][0])
G.set_pos(pos={centers[0]:[-0.3,0],centers[1]:[0.3,0],
left[0]:[-0.14,0.15], right[0]:[0.14,0.15],
left[1]:[-0.14,-0.15],right[1]:[0.14,-0.15],
left[2]:[-0.14,-0.3],right[2]:[0.14,-0.3]})
G.relabel(range(1,n+1))
return G
def make_graph(M):
"""
Code extracted from Sage's elliptic curve isogeny class (reshaped
in the case maxdegree==12)
"""
from sage.schemes.elliptic_curves.ell_curve_isogeny import fill_isogeny_matrix, unfill_isogeny_matrix
from sage.graphs.graph import Graph
n = M.nrows() # = M.ncols()
G = Graph(unfill_isogeny_matrix(M), format='weighted_adjacency_matrix')
MM = fill_isogeny_matrix(M)
# The maximum degree classifies the shape of the isogeny
# graph, though the number of vertices is often enough.
# This only holds over Q, so this code will need to change
# once other isogeny classes are implemented.
if n == 1:
# one vertex
pass
elif n == 2:
# one edge, two vertices. We align horizontally and put
# the lower number on the left vertex.
G.set_pos(pos={0:[-0.5,0],1:[0.5,0]})
else:
maxdegree = max(max(MM))
if n == 3:
# o--o--o
centervert = [i for i in range(3) if max(MM.row(i)) < maxdegree][0]
other = [i for i in range(3) if i != centervert]
G.set_pos(pos={centervert:[0,0],other[0]:[-1,0],other[1]:[1,0]})
elif maxdegree == 4:
# o--o<8
centervert = [i for i in range(4) if max(MM.row(i)) < maxdegree][0]
other = [i for i in range(4) if i != centervert]
G.set_pos(pos={centervert:[0,0],other[0]:[0,1],other[1]:[-0.8660254,-0.5],other[2]:[0.8660254,-0.5]})
elif maxdegree == 27:
# o--o--o--o
centers = [i for i in range(4) if list(MM.row(i)).count(3) == 2]
left = [j for j in range(4) if MM[centers[0],j] == 3 and j not in centers][0]
right = [j for j in range(4) if MM[centers[1],j] == 3 and j not in centers][0]
G.set_pos(pos={left:[-1.5,0],centers[0]:[-0.5,0],centers[1]:[0.5,0],right:[1.5,0]})
elif n == 4:
# square
opp = [i for i in range(1,4) if not MM[0,i].is_prime()][0]
other = [i for i in range(1,4) if i != opp]
G.set_pos(pos={0:[1,1],other[0]:[-1,1],opp:[-1,-1],other[1]:[1,-1]})
elif maxdegree == 8:
# 8>o--o<8
centers = [i for i in range(6) if list(MM.row(i)).count(2) == 3]
left = [j for j in range(6) if MM[centers[0],j] == 2 and j not in centers]
right = [j for j in range(6) if MM[centers[1],j] == 2 and j not in centers]
G.set_pos(pos={centers[0]:[-0.5,0],left[0]:[-1,0.8660254],left[1]:[-1,-0.8660254],centers[1]:[0.5,0],right[0]:[1,0.8660254],right[1]:[1,-0.8660254]})
elif maxdegree == 18:
# two squares joined on an edge
centers = [i for i in range(6) if list(MM.row(i)).count(3) == 2]
top = [j for j in range(6) if MM[centers[0],j] == 3]
bl = [j for j in range(6) if MM[top[0],j] == 2][0]
br = [j for j in range(6) if MM[top[1],j] == 2][0]
G.set_pos(pos={centers[0]:[0,0.5],centers[1]:[0,-0.5],top[0]:[-1,0.5],top[1]:[1,0.5],bl:[-1,-0.5],br:[1,-0.5]})
elif maxdegree == 16:
# tree from bottom, 3 regular except for the leaves.
centers = [i for i in range(8) if list(MM.row(i)).count(2) == 3]
center = [i for i in centers if len([j for j in centers if MM[i,j] == 2]) == 2][0]
centers.remove(center)
bottom = [j for j in range(8) if MM[center,j] == 2 and j not in centers][0]
left = [j for j in range(8) if MM[centers[0],j] == 2 and j != center]
right = [j for j in range(8) if MM[centers[1],j] == 2 and j != center]
G.set_pos(pos={center:[0,0],bottom:[0,-1],centers[0]:[-0.8660254,0.5],centers[1]:[0.8660254,0.5],left[0]:[-0.8660254,1.5],right[0]:[0.8660254,1.5],left[1]:[-1.7320508,0],right[1]:[1.7320508,0]})
elif maxdegree == 12:
# tent
centers = [i for i in range(8) if list(MM.row(i)).count(2) == 3]
left = [j for j in range(8) if MM[centers[0],j] == 2]
right = []
for i in range(3):
right.append([j for j in range(8) if MM[centers[1],j] == 2 and MM[left[i],j] == 3][0])
G.set_pos(pos={centers[0]:[-0.3,0],centers[1]:[0.3,0],
left[0]:[-0.14,0.15], right[0]:[0.14,0.15],
left[1]:[-0.14,-0.15],right[1]:[0.14,-0.15],
left[2]:[-0.14,-0.3],right[2]:[0.14,-0.3]})
G.relabel(range(1,n+1))
return G
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)
