The aim of this paper is to study certain family of elliptic curves $\{ \mathscr{ X}_H\}_H$ defined over a number field $F$ arising from hyperplane sections of some cubic surface $\mathscr{ X}/F$ associated to a cyclic cubic extension $K/F$. We show that each $\mathscr{ X}_H$ admits a 3-isogeny $\phi$ over $F$ and the dual Selmer group $S^{ (\hat{ \phi})}(\hat{ \mathscr{ X}_H}/F)$ is bounded by a kind of unit/class groups attached to $K/F$. This is proven via certain rational function on the elliptic curve $\mathscr{ X}_H$ with nice property. We also prove that the Shafarevich-Tate group $\text{ \cyr X} (\hat{ \mathscr{ X}_H}/\rat)[\hat{ \phi}]$ coincides with a class group of $K$ as a special case.
Categories: math.NT