Behaviour of the order of Tate-Shafarevich groups for the quadratic twists of elliptic curves

We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve $E$ over $\mathbb{Q}$ and positive integer $k$, an asymptotic formula for the number of quadratic twists $E_d$, $d$ positive square-free integers less than $X$, with finite group $E_d(\Bbb Q)$and$|\Sha(E_d(\Bbb Q))| = k^2$. This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve $X_0(49)$. In section 8 we exhibit $88$ examples of rank zero elliptic curves with $\mathrm{\mid }\text{Sha}(E)\mathrm{\mid }>{63408}^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $\mathrm{\mid }\text{Sha}(E)\mathrm{\mid }={1029212}^2$.