There are a couple small issues with graphing, but the functions are properly computing isogenies and graphing most of them! See images below.
x = var('x')
K.<a> = NumberField(x^2-x-1)
import psage.modform.hilbert.sqrt5.tables as sqrt5
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def ap(E,p):
return E.change_ring(p.residue_field()).trace_of_frobenius()
R.<ch> = GF(2)[]
def frob(E,p):
t = ap(E,p)
return ch^2 - ap(E, p)*ch + int(p.norm())
def disc(E, p):
t = ap(E, p)
return t^2 - 4*p.norm()
def isogeny_primes(E, norm_bound, isog_degree_bound):
P = [p for p in sqrt5.ideals_of_bounded_norm(norm_bound) if p.is_prime() and E.has_good_reduction(p)]
w = set(primes(isog_degree_bound+1))
i = 0
w.remove(2)
while len(w) > 0 and i < len(P):
d = disc(E, P[i])
w = [ell for ell in w if not (legendre_symbol(d,ell) == -1)]
i = i +1
i = 0
while i < len(P):
if frob(E,P[i]).is_irreducible():
break
i = i+1
if i == len(P):
w.insert(0,2)
return w
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def isogeny_class_computation(E,p):
if p != 2:
E = E.short_weierstrass_model()
F = E.division_polynomial(p).change_ring(K)
f = F.factor()
i=0
S.<x> = K[]
t=f[0][0].degree()
w=[S(1)]
while t < (p+1)/2:
for j in w:
if (j*f[i][0]).degree() < (p + 1)/2 and (j*f[i][0]).divides(F):
w.append(j*f[i][0])
t = f[i+1][0].degree()
i+=1
v=[]
o = 1
while o < len(w):
try:
v.append(E.isogeny(w[o]).codomain())
except ValueError:
pass
o+=1
v = [F.change_ring(K).global_minimal_model() for F in v]
return v
else:
w = [Q for Q in E.torsion_subgroup() if order(Q)==2]
v = [E.isogeny(E(Q)).codomain() for Q in w]
return v
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#def isogeny_class_computation2(E,p): #William's
# if p != 2:
# E = E.short_weierstrass_model()
# F = E.division_polynomial(p)
# print F.parent(), type(F), F
# F = F.change_ring(K)
# try:
# w = (f for f in F.divisors() if f.degree() == (p-1)/2)
# except AttributeError:
# print F.parent(), type(F)
# raise AttributeError
# for f in w:
# try:
# v.append(E.change_ring(K).isogeny(f).codomain())
# except ValueError:
# pass
# v = [F.change_ring(K).global_minimal_model() for F in v]
# return v
# else:
# w = [Q for Q in E.torsion_subgroup() if order(Q)==2]
# v = [E.isogeny(E(Q)).codomain() for Q in w]
# return v
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def curve_isogeny_classes(E):
isolist = isogeny_primes(E,1000,1000)
for i in isolist:
print i,'- isogeny class: '
for j in isogeny_class_computation(E,i):
print ' ',j.a_invariants()
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def curve_isogeny_vector(E):
curve_list = [E]
i = 0
Adj = matrix(50)
ins = 1
while i < len(curve_list):
isolist = isogeny_primes(curve_list[i],500,500)
for j in isolist:
for F in isogeny_class_computation(curve_list[i],j):
bool = True
for G in curve_list:
if F.is_isomorphic(G):
bool = False
Adj[i,curve_list.index(G)]=j
Adj[curve_list.index(G),i]=j
if bool:
curve_list.append(F)
Adj[i,ins]=j
Adj[ins,i]=j
ins += 1
i+=1
Adj = Adj.submatrix(nrows=len(curve_list),ncols=len(curve_list))
return curve_list, Adj
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def isogeny_printer(E):
X,Y=curve_isogeny_vector(E)
counter = 0
for i in X:
print 'curve',counter,': ',i.a_invariants()
counter +=1
show(Y)
show(Graph(Y))
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E = EllipticCurve(K,[a,-a+1,1,-1,0])
isogeny_printer(E)
curve 0 : (a, -a + 1, 1, -1, 0) curve 1 : (a, -a + 1, 1, -5*a + 9, -8*a + 6) curve 2 : (a, -a + 1, 1, -45*a - 151, -264*a - 858) curve 3 : (a, -a + 1, 1, -3365*a - 12621, -229016*a - 564210) curve 0 : (a, -a + 1, 1, -1, 0) curve 1 : (a, -a + 1, 1, -5*a + 9, -8*a + 6) curve 2 : (a, -a + 1, 1, -45*a - 151, -264*a - 858) curve 3 : (a, -a + 1, 1, -3365*a - 12621, -229016*a - 564210) |
F = EllipticCurve(K,[0,a,1,1,0])
isogeny_printer(F)
curve 0 : (0, a, 1, 1, 0) curve 1 : (0, a, 1, 30*a - 59, 102*a - 186) curve 0 : (0, a, 1, 1, 0) curve 1 : (0, a, 1, 30*a - 59, 102*a - 186) |
G = EllipticCurve(K,[1,-1,a,-2*a,a])
isogeny_printer(G)
curve 0 : (1, -1, a, -2*a, a) curve 1 : (1, -1, a, -17*a - 15, 52*a + 30) curve 2 : (1, -1, a, 13*a - 15, 20*a - 26) curve 3 : (1, -1, a + 1, a - 2, -2*a + 1) curve 4 : (1, -1, a, 148*a - 285, 1262*a - 1943) curve 5 : (1, -1, a + 1, 16*a - 32, -53*a + 82) curve 6 : (1, -1, a + 1, -14*a - 2, -21*a - 6) curve 7 : (1, -1, a + 1, -149*a - 137, -1263*a - 681) curve 0 : (1, -1, a, -2*a, a) curve 1 : (1, -1, a, -17*a - 15, 52*a + 30) curve 2 : (1, -1, a, 13*a - 15, 20*a - 26) curve 3 : (1, -1, a + 1, a - 2, -2*a + 1) curve 4 : (1, -1, a, 148*a - 285, 1262*a - 1943) curve 5 : (1, -1, a + 1, 16*a - 32, -53*a + 82) curve 6 : (1, -1, a + 1, -14*a - 2, -21*a - 6) curve 7 : (1, -1, a + 1, -149*a - 137, -1263*a - 681) |
H = EllipticCurve(K,[1,-1,1,-5,-7])
isogeny_printer(H)
curve 0 : (1, -1, 1, -5, -7) curve 1 : (1, -1, 1, -95, -331) curve 2 : (1, -1, 1, 40, 155) curve 3 : (1, -1, 1, -320, 1883) curve 4 : (1, -1, 1, -1535, 23591) curve 5 : (1, -1, 1, -24575, 1488935) curve 0 : (1, -1, 1, -5, -7) curve 1 : (1, -1, 1, -95, -331) curve 2 : (1, -1, 1, 40, 155) curve 3 : (1, -1, 1, -320, 1883) curve 4 : (1, -1, 1, -1535, 23591) curve 5 : (1, -1, 1, -24575, 1488935) |
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