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For any number field K with a complex place, we present an infinite family ofelliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim\mathbb{F}_2 E^F(K)[2] + r_2$ for every quadratic twist E^F of every curve E inthis family, where r_2 is the number of complex places of K. This provides acounterexample to a conjecture appearing in work of Mazur and Rubin.

A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists