The Selmer groups and the ambiguous ideal class groups of cubic fields

In this paper, we study a family of elliptic curves with CM by ℚ(√-3) which also admits a ℚ-rational isogeny of degree 3. We find a relation between the Selmer groups of the elliptic curves and the ambiguous ideal class groups of certain cubic fields . We also find some bounds for the dimension of the 3-Selmer group over ℚ, whose upper bound is also an upper bound of the rank of the elliptic curve.