Non-trivial Shafarevich-Tate Groups of Elliptic Curves

We characterize quadratic twists of $y^2=x(x-a^2)(x+b^2)$ with Mordell-Weil groups and $2$-primary part of Shafarevich-Tate groups being isomorphic to $(\text{mathb}Z\mathrm{/}2\mathbb{Z}{)}^2$ under certain conditions. We also obtain the distribution result of these elliptic curves.