Thue equations and torsion groups of elliptic curves (Q999716)

From the text: A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations. Theorem. Given an elliptic curve \(E: Y^2=X^3+AX+B\) with \(A,B\in\mathbb Z\), for \(n\in\{5,7,8,9\}\) there are homogeneous binary polynomials \(F_n, G_n\in\mathbb Z[p,q]\), at least one of them irreducible, such that \(E\) has a rational point of order \(n\) if and only if there is a solution to the system \[ 6^4A=F_n,\quad 6^6B=G_n. \]