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 E-n : y(2) = x(3) - 27n(2 )and a dual isogeny phi : E-n -> E-n. We show that for almost all n, the rank of Sel(phi)(E-n) is 0, and the rank of Sel(phi)(E-n) is determined by the number of prime factors of n that are congruent to 2 mod 3 and the congruence class of n mod 9.