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

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 ${\widehat{E}}_n:y^2=x^3-27n^2$ and a dual isogeny $\widehat{\phi }:{\widehat{E}}_n\to E_n$. We show that for almost all $n$, the rank of ${\mathrm{S}\mathrm{e}\mathrm{l}}_{\phi }(E_n)$ is $0$, and the rank of ${\mathrm{S}\mathrm{e}\mathrm{l}}_{\widehat{\phi }}({\widehat{E}}_n)$ is determined by the number of prime factors of $n$ that are congruent to $23$ and the congruence class of $n9$.