The 3-Isogeny Selmer Groups of the Elliptic Curves y2=x3+n2.

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$|⁠ , the rank of |$\operatorname {Sel}_{\phi }(E_{n})$| is |$0$|⁠ , and the rank of |$\operatorname {Sel}_{\hat {\phi }}(\hat {E}_{n})$| is determined by the number of prime factors of |$n$| that are congruent to |$2\bmod 3$| and the congruence class of |$n\bmod 9$|⁠. [ABSTRACT FROM AUTHOR]