2016-02-12-homework-6.sagews

Math 582: computational number theory

Homework 6 -- due by Monday, Feb 22 at 11am

Problem 1.

For each of the possible torsion subgroups of elliptic curves over $\QQ$ (according to Mazur's theorem), find an elliptic curves of conductor at least 1000 that has that torsion subgroup.

# C1 -> 1001c1
# C2 -> 1001b3
# C3 -> 1005b1
# C4 -> 1008e4
# C5 -> 1050o1
# C6 -> 1110h2
# C7 -> 1230k1
# C8 -> 1230f1
# C9 -> 1482l1
# C10 -> 6270r2
# C12 -> 2730bd1
# C2 x C2 -> 1001b2
# C2 x C4 -> 1110k2
# C2 x C6 -> 2310g2
# C2 x C8 -> 46410cn2
for label in [
    "1001c1",
    "1001b3",
    "1005b1",
    "1008e4",
    "1050o1",
    "1110h2",
    "1230k1",
    "1230f1",
    "1482l1",
    "6270r2",
    "2730bd1",
    "1001b2",
    "1110k2",
    "2310g2",
    "46410cn2",
]:
    E = EllipticCurve(label)
    print E.torsion_subgroup()

Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2 + y = x^3 - 199*x + 1092 over Rational Field Torsion Subgroup isomorphic to Z/2 associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 9916*x - 377564 over Rational Field Torsion Subgroup isomorphic to Z/3 associated to the Elliptic Curve defined by y^2 + y = x^3 + x^2 + 239*x + 295 over Rational Field Torsion Subgroup isomorphic to Z/4 associated to the Elliptic Curve defined by y^2 = x^3 - 36291*x + 2661010 over Rational Field Torsion Subgroup isomorphic to Z/5 associated to the Elliptic Curve defined by y^2 + x*y = x^3 + 22*x - 2748 over Rational Field Torsion Subgroup isomorphic to Z/6 associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 - 5313*x + 148588 over Rational Field Torsion Subgroup isomorphic to Z/7 associated to the Elliptic Curve defined by y^2 + x*y = x^3 + 2305*x - 15975 over Rational Field Torsion Subgroup isomorphic to Z/8 associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 4890*x + 129447 over Rational Field Torsion Subgroup isomorphic to Z/9 associated to the Elliptic Curve defined by y^2 + x*y = x^3 + 12948*x + 421776 over Rational Field Torsion Subgroup isomorphic to Z/10 associated to the Elliptic Curve defined by y^2 + x*y = x^3 - 454955*x + 118072977 over Rational Field Torsion Subgroup isomorphic to Z/12 associated to the Elliptic Curve defined by y^2 + x*y = x^3 - 25725*x + 1577457 over Rational Field Torsion Subgroup isomorphic to Z/2 + Z/2 associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 621*x - 5764 over Rational Field Torsion Subgroup isomorphic to Z/4 + Z/2 associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 21325*x + 1187267 over Rational Field Torsion Subgroup isomorphic to Z/6 + Z/2 associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 - 286854*x + 58872856 over Rational Field Torsion Subgroup isomorphic to Z/8 + Z/2 associated to the Elliptic Curve defined by y^2 + x*y = x^3 - 8696090*x + 9838496100 over Rational Field

︠835e4c29-f0a9-479f-a538-168c832ca1ad︠

︠49c0e624-11fc-41d2-8733-66a12a734f92︠

︠85d7bd6a-c70f-4ca5-9512-0e03b0cf1a52︠

︠a41a2410-339b-4534-8a18-35243b408181i︠
%md
## Problem 2.
Compute the rank and the size of the isogeny class of each curve that you found in problem 1.

Problem 2.

Compute the rank and the size of the isogeny class of each curve that you found in problem 1.

for label in [
    "1001c1",
    "1001b3",
    "1005b1",
    "1008e4",
    "1050o1",
    "1110h2",
    "1230k1",
    "1230f1",
    "1482l1",
    "6270r2",
    "2730bd1",
    "1001b2",
    "1110k2",
    "2310g2",
    "46410cn2",
]:
    E = EllipticCurve(label)
    print "%s   \t--->\trank=%d\tisogenyclass=%d" % (label,E.rank(),len(E.isogeny_graph().vertices()))

1001c1 ---> rank=2 isogenyclass=1 1001b3 ---> rank=0 isogenyclass=4 1005b1 ---> rank=1 isogenyclass=2 1008e4 ---> rank=0 isogenyclass=4 1050o1 ---> rank=0 isogenyclass=2 1110h2 ---> rank=0 isogenyclass=4 1230k1 ---> rank=0 isogenyclass=2 1230f1 ---> rank=0 isogenyclass=8 1482l1 ---> rank=1 isogenyclass=3 6270r2 ---> rank=1 isogenyclass=4 2730bd1 ---> rank=0 isogenyclass=8 1001b2 ---> rank=0 isogenyclass=4 1110k2 ---> rank=0 isogenyclass=6 2310g2 ---> rank=1 isogenyclass=8 46410cn2 ---> rank=0 isogenyclass=8

︠1eee3947-5b13-46e6-a1dc-56de22bea533︠

︠31942c98-ddc1-42d7-bf4e-293f357b4f65︠

︠290caff8-2f56-4cce-a57a-808c14366c95︠

︠92b8e8a2-4acc-40de-810c-49d80d2b8bb5︠

︠b8b25ecc-15ee-4ebd-b26d-c8d5518ad9e0i︠
%md
## Problem 3.
By brute force search (or whatever), find an elliptic curve $E$ over a finite field $\FF_p$ such that $E(\FF_p)$ has order divisible by $2016$.

Problem 3.

By brute force search (or whatever), find an elliptic curve $E$ over a finite field $\FF_p$ such that $E(\FF_p)$ has order divisible by $2016$.

# squarefree part of the discriminant of x^2 + 2x + 2017 is -14, #fun fact: 2016 + 1 is prime!
F.<isqrt14> = QuadraticField(-14)
K.<a> = F.extension(F.hilbert_class_polynomial())
EK = EllipticCurve_from_j(a)
P = K.factor(2017)[0][0]
E = EK.reduction(P)
E.count_points()

2016

## NICE