We consider a family of genus one curves defined by

$$displaystyle d_0 z^2 = F_{a,b}(u,v) text{ (1)},$$

where

$$F_{a,b}(u,v) = a(u^2 – v^2)^2 + 4bu^2 v^2$$

and $d_0 = F_{a,b}(u_0, v_0)$, where $u_0, v_0$ are fixed integers independent of $a,b$. Then, provided that $a,b$ are integers, the curve defined by (1) is a genus one curve with a marked rational point, namely $(u_0, v_0, 1)$.

How does one determine whether the marked point $(u_0, v_0, 1)$ is torsion? As $a,b$ vary over a box, say $max{|a|, |b|} leq X$, can one say anything about the distribution of pairs $(a,b)$ for which $(u_0, v_0, 1)$ is torsion/not torsion?