Elliptic curves related to cyclic cubic extensions

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