Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups: Distribution Part

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We study the distribution of a subclass congruent elliptic curve E⁽ⁿ⁾: y²=x³-n²x, where n is congruent to 1 8 with all prime factors congruent to 1 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 (Z /2Z)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 (Z /2Z)⁴.