On the quadratic twist of elliptic curves with full 2-torsion

Let [Formula: see text] be an elliptic curve with full [Formula: see text]-torsion group, where [Formula: see text] and [Formula: see text] are coprime integers and [Formula: see text] is a square. Assume that the [Formula: see text]-Selmer group of [Formula: see text] has rank two. We characterize all quadratic twists of [Formula: see text] with Mordell-Weil rank zero and [Formula: see text]-primary Shafarevich-Tate groups [Formula: see text], under certain conditions. We also obtain a distribution result for these elliptic curves.