Let K be a number field and E/K be an elliptic curve with no 2-torsion points. In the present article we give lower and upper bounds for the 2-Selmer rank of E in terms of the 2-torsion of a narrow class group of a certain cubic extension of K attached to E. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of E have the same 2-Selmer group.
