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
