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 $\big(\mathbb Z /2\mathbb Z\big)^{ 4}$.
Categories: math.NT