  We study the distribution of a subclass congruent elliptic curve $E^{(n)}:
y^2=x^3-n^2x$, where $n$ is congruent to $1\pmod 8$ with all prime factors
congruent to $1\pmod 4$. We prove an independence of residue symbol property.
Consequently we get the distribution of rank zero such $E^{(n)}$ with
$2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z
/2\mathbb Z\big)^2$. We also obtain a lower bound of the number of such
$E^{(n)}$ with rank zero and $2$-primary part of Shafarevich-Tate group
isomorphic to $\big(\mathbb Z /2\mathbb Z\big)^{4}$.
