New sequences of non-free rational points

We exhibit some new infinite families of rational values of $\tau$, some of them squares of rationals, for which the group or even the semigroup generated by the matrices $\left( \begin{smallmatrix} 1 & 1 \ 0 & 1 \end{smallmatrix} \right)$and$\left( \begin{smallmatrix} 1 & 0 \ \tau & 1 \end{smallmatrix} \right)$ is not free.