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We give an upper and a lower bound on the rank of the $\\varphi$-Selmer group of $E_a$ over $\\mathbb Q(\\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\\mathbb Q(\\zeta_3)$. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. 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