Level structure of a modular curve (awaiting review)

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 $\GL_2(\Z/N\Z)$.