  Record values are determined for the order $|\Sha|$ of the Tate--Shafarevich
group of an elliptic curve $E$, computed analytically by the
Birch--Swinnerton-Dyer conjecture, and for the Goldfeld--Szpiro ratio
$G=|\Sha|/\sqrt{N}$, where $N$ is the conductor of $E$. The curves have rank
zero and are isogenous to quadratic twists of Frey curves constructed from
coprime positive integers $(a,b,c)$ with $a+b=c$ and $c>r^{1.4}$, where the
radical $r$ is the product of the primes dividing $abc$. Curves with
$|\Sha|>250000^2$ and $G>12$ are found in 20 isogeny classes. Three curves have
$G>150$. The largest value of $|\Sha|$ is $1937832^2>3.755\times10^{12}$. This
is more than 3.5 times the previous record, which had been computed at a cost
about 600 times greater than that for the new record. The primes 25913, 27457,
36929 and 49253 are identified as divisors of $|\Sha|$ values.
