Level structure of a modular curve (awaiting review)

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