  Let $n$ be a positive square-free integer, where every odd prime factor of
$n$ has form $8a\pm 1$. We determine when $n$ is non-congruent with second
minimal $2$-primary Shafarevich-Tate group, in terms of the $4$-ranks of class
groups and a Jacobi symbol. In particular, when every odd prime factor of $n$
has form $8a+1$, this condition is equivalent to the vanishing of the $4$-rank
of the tame kernel of $\mathbb Q(\sqrt{n})$ for odd $n$, or $\mathbb
Q(\sqrt{-n})$ for even $n$. This generalizes previous results.
