Note on the rank of quadratic twists of Mordell equations

Let E be the elliptic curve given by a Mordell equation y2=x3−A where A∈Z. Michael Stoll found a precise formula for the size of a Selmer group of E for certain values of A. For D∈Z, let ED denote the quadratic twist Dy2=x3−A. We use Stoll's formula to show that for a positive square-free integer A≡1 or and for a nonnegative integer k, we can compute a lower bound for the proportion of square-free integers D up to X such that rankED(Q)⩽2k. We also compute an upper bound for a certain average rank of quadratic twists of E.