when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2=(x^2-A_{1}^2)(x^2-A_{2}^2)(x^2-A_{3}^2) ,A_{i}\in \mathbb{Q},i=1,2,3$$

I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that $A_{i}=\dfrac{p_{i}}{q_{i}},i=1,2,3$,can we estimate upper bound of $p_{i}$ or $q_{i}?$