Optimal curves differing by a 5-isogeny

Dongho Byeon¹ ; Taekyung Kim¹

¹ Department of Mathematics Seoul National University Seoul, Korea

Acta Arithmetica, Tome 165 (2014) no. 4, pp. 351-359.

Résumé

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.

Détail

DOI : 10.4064/aa165-4-5

Keywords: optimal curve isogeny class mathcal elliptic curves defined mathbb conductor stein watkins conjectured differ isogeny only paper conjecture square free divisible other hadano conjectured elliptic curve defined mathbb rational point order isogenous curve langle rangle has rational point order again only process proof stein watkinss conjecture hadanos conjecture