Rational $D(q)$-quadruples

Abstract: For a rational number $q$, a rational $D(q)$-$n$-tuple is a set of $n$ distinct nonzero rationals ${a_1, a_2, \dots, a_n}$ such that $a_ia_j+q$ is a rational square for all $1 \leqslant i < j \leqslant n$. For every $q$ we find all rational $m$ such that there exists a $D(q)$-quadruple with product $abcd=m$. We describe all such quadruples using points on a specific elliptic curve depending on $(q,m).$