Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and E(F_q) be an elliptic curve over F_q given by the equation y²=x³+Ax+B. For a positive integer n we denote by #E(F_qⁿ) the number of rational points on E (including infinity) over the extension F_qⁿ. Under a mild technical condition, we show that the sequence {#E(F_qⁿ)}_n>0 contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then #E(F_qⁿ) is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range q<50 and n≤1000.