Congruent elliptic curves with non-trivial Shafarevich-Tate groups

We study 2-primary parts Ш(E⁽ⁿ⁾/Q)2^∞] of Shafarevich-Tate groups of congruent elliptic curves E⁽ⁿ⁾: y² = x³?n²x, n ∈ Q^×/Q^×2. Previous results focused on finding sufficient conditions for Ш(E⁽ⁿ⁾/Q)2^∞] trivial or isomorphic to (Z/2Z)². Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E⁽ⁿ⁾ is isomorphic to (Z/2Z)² and the Mordell-Weil rank of E⁽ⁿ⁾ is zero, provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for Ш(E⁽ⁿ⁾/Q)2^∞] ? (Z/2Z)^2k, where k ≥ 2.