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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   $n\bmod 9$  .

The 3-Isogeny Selmer Groups of the Elliptic Curves y 2 = x 3 + n 2