The groups ${\mathrm{SL}}_2({\mathbb{F}}_{13})$ and ${\mathrm{SL}}_2({\mathbb{F}}_{19})$ are Galois over $\mathbb{Q}$

In this paper, we show that ${\mathrm{SL}}_2({\mathbb{F}}_{13})$ and ${\mathrm{SL}}_2({\mathbb{F}}_{19})$ are Galois groups of totally real extensions of $\mathbb{Q}$. For each of these primes $\mathrm{\ell }$, we find a polynomial of degree $\mathrm{\ell }+1$ whose splitting field has Galois group ${\mathrm{PSL}}_2({\mathbb{F}}_{\mathrm{\ell }})$. These fields satisfy Böge’s criterion and the associated central embedding problem has a proper solution with Galois group ${\mathrm{SL}}_2({\mathbb{F}}_{\mathrm{\ell }})$. One can choose the resulting fields totally real via a quadratic twist. The degree $14$ polynomial is found through a genus one Hurwitz family of degree 14 covers. The degree 20 polynomial is found using Yang’s modular equations for the Shimura curve $X_6^{\ast }(1)$.