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.

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- Review status: beta
- Last edited by Andrew Sutherland on 2023-07-09 09:04:41

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- 2023-07-09 09:04:41 by Andrew Sutherland
- 2023-01-25 20:08:56 by Andrew Sutherland
- 2023-01-25 18:49:09 by Andrew Sutherland
- 2023-01-25 18:47:55 by Andrew Sutherland
- 2023-01-25 18:41:22 by Andrew Sutherland
- 2022-11-06 17:05:45 by Ciaran Schembri

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