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.


︠3d3a45b3-1d17-417e-8a5e-c3449388c5ae︠

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

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

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

︠a41a2410-339b-4534-8a18-35243b408181i︠
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## 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.


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︠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
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## 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 $2016$ .