  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.
