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 |ш(E)|>634082, which was the largest previously known value for any explicit curve. Our record is an elliptic curve E with |ш(E)|=10292122=24⋅792⋅32572. We use deep results by Kolyvagin, Kato, Skinner–Urban and Skinner to prove that, in some cases, these orders are the true orders of ш. For instance, 4105362 is the true order of ш(E) for E=E4(21,−233) from the table in Section 2.3.