Rank Frequencies for Quadratic Twists of Elliptic Curves

We give explicit examples of infinite families of elliptic curves E over Q with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We recover some results announced by Mestre, as well as some additional families. Suppose D is a squarefree integer and let rE(D) denote the rank of the quadratic twist of E by D. We apply results of Stewart and Top to our examples to obtain results of the form #{D : |D| < x, rE(D) ≥ 2} x #{D : |D| < x, rE(D) ≥ 3} x for all sufficiently large x.