Elliptic curves with exceptionally large analytic order of the Tate-Shafarevich groups

We exhibit $88$ examples of rank zero elliptic curves over the rationals with $\mathrm{\mid }\text{sza}(E)\mathrm{\mid }>{63408}^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|\sza(E)| = 1029212^2 = 2^4\cdot 79^2 \cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of $\text{sza}$. For instance, ${410536}^2$ is the true order of $\text{sza}(E)$ for $E=E_4(21,-233)$ from the table in section 2.3.