Isogeny comp -- Sage

x = var('x') K.<a> = NumberField(x^2-x-1) import psage.modform.hilbert.sqrt5.tables as sqrt5

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

def isogeny_class_computation(E,p): if p != 2: E = E.short_weierstrass_model() F = E.division_polynomial(p).change_ring(K) v = [] for f in [f for f in divisors(F) if f.degree() == (p-1)/2]: 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

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()

E = EllipticCurve([1,a+1,a,a,0] ); E.conductor().norm()

E.torsion_subgroup()

Torsion Subgroup isomorphic to Z/8 associated to the Elliptic Curve
defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in
a with defining polynomial x^2 - x - 1

Torsion Subgroup isomorphic to Z/8 associated to the Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1

time curve_isogeny_classes(E)

2 - isogeny class: 
   (1, a + 1, a, a, 0)
   (1, a + 1, a, a - 5, 3*a - 5)
Time: CPU 25.08 s, Wall: 25.08 s

2 - isogeny class: 
   (1, a + 1, a, a, 0)
   (1, a + 1, a, a - 5, 3*a - 5)
Time: CPU 25.08 s, Wall: 25.08 s

isogeny_class_computation(E,2)

[Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over
Number Field in a with defining polynomial x^2 - x - 1, Elliptic Curve
defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + (a-5)*x + (3*a-5) over
Number Field in a with defining polynomial x^2 - x - 1]

[Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1, Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + (a-5)*x + (3*a-5) over Number Field in a with defining polynomial x^2 - x - 1]

E = EllipticCurve(K,[1,0,1,-1,-2]); E

Elliptic Curve defined by y^2 + x*y + y = x^3 + (-1)*x + (-2) over
Number Field in a with defining polynomial x^2 - x - 1

Elliptic Curve defined by y^2 + x*y + y = x^3 + (-1)*x + (-2) over Number Field in a with defining polynomial x^2 - x - 1

for F in isogeny_class_computation(E,3): print F.a_invariants()

(1, 0, 1, -126, -552)

(1, 0, 1, -126, -552)

for F in isogeny_class_computation(E,5): print F.a_invariants()

(1, 1, 1, -3, 1)

(1, 1, 1, -3, 1)

for E in cremona_optimal_curves([37]): print E.rank(), E.quadratic_twist(5).rank()

1 0
0 1

1 0
0 1

(163^2-1)/2

sage.schemes.elliptic_curves.ell_curve_isogeny.isogenies_sporadic_Q(163)

Traceback (click to the left of this block for traceback)
...
ValueError: The elliptic curve must be defined over QQ.

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_29.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("c2FnZS5zY2hlbWVzLmVsbGlwdGljX2N1cnZlcy5lbGxfY3VydmVfaXNvZ2VueS5pc29nZW5pZXNfc3BvcmFkaWNfUSgxNjMp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpdiiVVW/___code___.py", line 3, in <module>
    exec compile(u'sage.schemes.elliptic_curves.ell_curve_isogeny.isogenies_sporadic_Q(_sage_const_163 )
  File "", line 1, in <module>
    
  File "/sagenb/sage_install/sage-4.7/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_curve_isogeny.py", line 4131, in isogenies_sporadic_Q
    raise ValueError, "The elliptic curve must be defined over QQ."
ValueError: The elliptic curve must be defined over QQ.

E = EllipticCurve([-34790720, -78984748304])

E.conductor().factor()

2^4 * 163^2

2^4 * 163^2

F = E.minimal_quadratic_twist()[0]

F.conductor().factor()

163^2

163^2

F.isogenies_prime_degree()

[Isogeny of degree 163 from Elliptic Curve defined by y^2 + y = x^3 -
2174420*x + 1234136692 over Rational Field to Elliptic Curve defined by
y^2 + y = x^3 - 57772164980*x - 5344733777551611 over Rational Field]

[Isogeny of degree 163 from Elliptic Curve defined by y^2 + y = x^3 - 2174420*x + 1234136692 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - 57772164980*x - 5344733777551611 over Rational Field]

F.conductor()

163^4

705911761

705911761

Isogeny comp

332 days ago by WilliamStein