$X_{S_4}(\ell)$ (awaiting review)

For $\ell$ an odd prime, $X_{S_4}(\ell)$ is the modular curve $X_H$ for $H\le \GL_2(\widehat\Z)$ the inverse image of the subgroup of $\PGL_2(\Z/\ell\Z)$ isomorphic to $S_4$ (which is unique up to conjugacy). It parameterizes elliptic curves whose mod-$\ell$ Galois representation has projective image $S_4$, one of the three exceptional groups $A_4$, $A_5$, $S_4$ of $\PGL_2(\ell)$ that can arise as projective mod-$\ell$ images, and the only one that can arise for elliptic curves over $\Q$.

The subgroup $H$ contains $-I$ and has surjective determinant when $\ell \equiv \pm 3\bmod 8$, but not otherwise.