  We study a subclass of congruent elliptic curves $E^{(n)}: y^2=x^3-n^2x$,
where $n$ is a positive integer congruent to $1\pmod 8$ with all prime factors
congruent to $1\pmod 4$. We characterize such $E^{(n)}$ with Mordell-Weil rank
zero and $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb
Z/2\mathbb Z \big)^2$. We also discuss such $E^{(n)}$ with 2-primary part of
Shafarevich-Tate group isomorphic to $\big(\mathbb Z/2\mathbb Z \big)^{2k}$
with $k\ge2$.
