Let $H$ be an open subgroup of $\GL_2(\widehat\Z)$ of level $N$, let $\pi_N\colon \GL_2(\widehat\Z)\to \GL_2(\Z/N\Z)$ be the natural projection, and let $E$ be an elliptic curve over a number field $K$.

An **$H$-level structure** on $E$ is the $H$-orbit $[\iota]_H:=\{ h\circ \iota\colon h\in \pi_N(H)\}$ of an isomorphism $\iota\colon E[N]\overset{\sim}{\rightarrow}(\Z/N\Z)^2$.

An $H$-level structure on $E$ is **rational** if it lies in a $\Gal_K$-stable isomorphism class of pairs $(E,[\iota]_H)$, where $\sigma\in \Gal_K$ acts via $(E,[\iota]_H)\mapsto (E^\sigma,[\iota\circ\sigma^{-1}]_H)$.
Two pairs $(E,[\iota]_H)$ and $(E',[\iota']_H)$ are isomorphic if there is an isomorphism $\phi\colon E\to E'$ that induces an isomorphism $\phi_N\colon E[N]\to E'[N]$ for which $\phi_N^*([\iota']_H) = [\iota]_H$.

If $E$ admits a rational $H$-level structure $[\iota]_H$ then image of its adelic Galois representation $\rho_E\colon \Gal_K\to \GL_2(\widehat\Z)$ is conjugate to a subgroup of $H$ and the isomorphism class of $(E,[\iota]_H)$ is a non-cuspidal $K$-rational point on the modular curve $X_H$.

When $-1\in H$ every non-cuspidal $K$-rational point on $X_H$ arises in this way. When $-1\not\in H$ this is almost true, but there may be exceptions at points with $j(E)=0,1728$.

Invariants of a rational $H$-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 $E[N]\subseteq E(L)$; this is simply the cardinality of the reduction of $H$ to $\GL_2(\Z/N\Z)$.

**Knowl status:**

- Review status: beta
- Last edited by L Alexander Betts on 2024-04-15 18:33:10

**Referred to by:**

**History:**(expand/hide all)

- 2024-04-15 18:33:10 by L Alexander Betts
- 2024-04-15 18:30:41 by L Alexander Betts
- 2024-03-22 14:05:10 by L Alexander Betts
- 2024-03-14 18:06:54 by Andrew Sutherland
- 2024-03-13 15:43:25 by Bjorn Poonen
- 2023-07-09 07:22:00 by Andrew Sutherland
- 2023-07-09 07:21:49 by Andrew Sutherland
- 2023-07-09 07:21:14 by Andrew Sutherland
- 2023-07-09 07:20:55 by Andrew Sutherland
- 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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