  We exhibit $88$ examples of rank zero elliptic curves over the rationals with
$|\sza(E)| > 63408^2$, 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 $\sza$. For instance, $410536^2$ is the true order of $\sza(E)$
for $E= E_4(21,-233)$ from the table in section 2.3.
