Let $H$ be an open subgroup of $\GL_2(\widehat\Z)$. An elliptic curve $E$ over a number field $K$ is said to have **$H$-level structure** if the image of the adelic Galois representation $\rho_E\colon \Gal_K\to \GL_2(\widehat\Z)$ is conjugate to a subgroup of $H$.

This is equivalent to requiring that the mod-$N$ Galois representation $\rho_{E,N}\colon \Gal_K\to \GL_2(\Z/N\Z)$ given by the action of $\Gal_K$ on the $N$-torsion subgroup $E[N]$ has image conjugate to a subgroup of the reduction of $H$ modulo its level $N$. Thus to specify a level structure, it suffices to give generators for the reduction of $H$ to $\GL_2(\Z/N\Z)$.

Invariants of the level structure include:

**Cyclic $\boldsymbol{N}$-isogeny field degree**: the minimal degree of an extension $L/K$ over which the base change $E_L$ admits a rational cyclic isogeny of degree $N$; equivalently, the index of the largest subgroup of $H$ fixing a subgroup of $(\Z/N\Z)^2$ isomorphic to $\Z/N\Z$.**Cyclic $\boldsymbol{N}$-torsion field degree**: the minimal degree of an extension $L/K$ for which $E_L$ has a rational point of order $N$; equivalently, the index of the largest subgroup of $H$ that fixes a point of order $N$ in $(\Z/N\Z)^2$.**N-torsion field degree**the minimal degree of an extension $L/K$ for which $E_L$ has full $N$-level structure; this is simple the cardinality of the reduction of $H$ to $\GL_2(\Z/N\Z)$.

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- Review status: beta
- Last edited by Andrew Sutherland on 2022-03-20 22:29:03

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- 2022-03-20 22:29:03 by Andrew Sutherland
- 2022-03-20 22:28:32 by Andrew Sutherland
- 2022-03-20 22:28:16 by Andrew Sutherland
- 2022-03-20 22:27:53 by Andrew Sutherland
- 2022-03-20 22:27:31 by Andrew Sutherland
- 2022-03-20 22:26:45 by Andrew Sutherland
- 2022-03-20 22:26:12 by Andrew Sutherland
- 2022-03-20 21:02:53 by Andrew Sutherland
- 2022-03-20 20:51:36 by Andrew Sutherland
- 2022-03-20 20:48:45 by Andrew Sutherland

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