New sequences of non-free rational points

We exhibit some new infinite families of rational values of $τ$, 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 $\left( \begin{smallmatrix} 1 & 0 \\ τ& 1 \end{smallmatrix} \right)$ is not free.

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