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 E_D denote the quadratic twist D y² = x³ - A . We use Stoll's formula to show that for a positive square-free integer A ≡ 1 or 25 mod 36 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 rank E_D ( Q ) {less-than or slanted equal to} 2 k. We also compute an upper bound for a certain average rank of quadratic twists of E.