  Let $q\in\mathbb{C}$, let \[a=\begin{pmatrix} 1&0\\1&1\end{pmatrix},\quad
b_q=\begin{pmatrix} 1&q\\0&1\end{pmatrix},\] and let
$G_q<\mathrm{SL}_2(\mathbb{C})$ be the group generated by $a$ and $b_q$. In
this paper, we study the problem of determining when the group $G_q$ is not
free for $|q|<4$ rational. We give a robust computational criterion which
allows us to prove that if $q=s/r$ for $|s|\leq 27$ then $G_q$ is non-free,
with the possible exception of $s=24$. In this latter case, we prove that the
set of denominators $r\in\mathbb{N}$ for which $G_{24/r}$ is non-free has
natural density $1$. For a general numerator $s>27$, we prove that the lower
density of denominators $r\in \mathbb{N}$ for which $G_{s/r}$ is non-free has a
lower bound \[ 1- \left(1-\frac{11}{s}\right) \prod_{n=1}^\infty
\left(1-\frac{4}{s^{2^n-1}}\right). \] Finally, we show that for a fixed $s$,
there are arbitrarily long sequences of consecutive denominators $r$ such that
$G_{s/r}$ is non-free. The proofs of some of the results are computer assisted,
and Mathematica code has been provided together with suitable documentation.
