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Given an elliptic curve $E$ over a number field $F$ and an isogeny $\varphi$of $E$ defined over $F$, the study of the $\varphi$-Selmer group has a richhistory going back to the works of Cassels and the recent works of Bhargava etal. and Chao Li. Let $E/\mathbb Q$ be an elliptic curve with a rational$3$-isogeny. In this article, we give an upper bound and a lower bound of therank of the Selmer group of $E$ over $\mathbb Q(\zeta_3)$ induced by the$3$-isogeny in terms of the $3$-part of the ideal class group of certainquadratic extension of $\mathbb Q(\zeta_3)$. Using our bounds on the Selmergroups, we prove some cases of Sylvester's conjecture on the rational cube sumproblem and also exhibit infinitely many elliptic curves of arbitrary large$3$-Selmer rank over $\mathbb Q(\zeta_3)$. Our method also produces infinitelymany imaginary quadratic fields and biquadratic fields with non-trivial$3$-class groups.

$3$-Selmer group, ideal class groups and cube sum problem