  Let $E: y^2=x(x-a^2)(x+b^2)$ be an elliptic curve with full $2$-torsion
group, where $a$ and $b$ are coprime integers and $2(a^2+b^2)$ is a square.
Assume that the $2$-Selmer group of $E$ has rank two. We characterize all
quadratic twists of $E$ with Mordell-Weil rank zero and $2$-primary
Shafarevich-Tate groups $(\mathbb Z/2\mathbb Z)^2$, under certain conditions.
We also obtain a distribution result of these elliptic curves.
