Thurston construction mapping classes with minimal dilatation

Given a pair of filling curves $\alpha ,\beta$ on a surface of genus $g$ with $n$ punctures, we explicitly compute the mapping classes realizing the minimal dilatation over all the pseudo-Anosov maps given by the Thurston construction on $\alpha ,\beta$. We do so by solving for the minimal spectral radius in a congruence subgroup of ${\text{PSL}}_2(\mathbb{Z})$. We apply this result to realized lower bounds on intersection number between $\alpha$ and $\beta$ to give the minimal dilatation over any Thurston construction pA map on ${\mathrm{\Sigma }}_{g,n}$ given by a filling pair $\alpha \cup \beta$.