Optimal curves differing by a 5-isogeny
Volume 165 / 2014
Acta Arithmetica 165 (2014), 351-359
MSC: Primary 11G05; Secondary 14K02.
DOI: 10.4064/aa165-4-5
Abstract
For $i=0,1$, let $E_i$ be the $X_i(N)$-optimal curve of an isogeny class $\mathcal {C}$ of elliptic curves defined over $\mathbb Q$ of conductor $N$. Stein and Watkins conjectured that $E_0$ and $E_1$ differ by a 5-isogeny if and only if $E_0=X_0(11)$ and $E_1=X_1(11)$. In this paper, we show that this conjecture is true if $N$ is square-free and is not divisible by $5$. On the other hand, Hadano conjectured that for an elliptic curve $E$ defined over $\mathbb Q$ with a rational point $P$ of order 5, the 5-isogenous curve $E':=E/\langle P \rangle $ has a rational point of order 5 again if and only if $E'=X_0(11)$ and $E=X_1(11)$. In the process of the proof of Stein and Watkins's conjecture, we show that Hadano's conjecture is not true.
