The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$

Abstract

Consider the family of elliptic curves $E_n:y^2=x^3+n^2$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $\phi$ from $E_n$ to $\hat{E}_n:y^2=x^3-27n^2$ and a dual isogeny $\hat{\phi}:\hat{E}_n\rightarrow E_n$. We show that for almost all $n$, the rank of $\mathrm{Sel}_{\phi}(E_n)$ is $0$, and the rank of $\mathrm{Sel}_{\hat{\phi}}(\hat{E}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\bmod 3$ and the congruence class of $n\bmod 9$.