Label of a modular curve (awaiting review)

The label of a modular curve $X_H$ is the same as the label of the open subgroup $H\le \GL_2(\widehat\Z)$. When $\det(H)=\widehat\Z^\times$ and $-I\in H$ this label has the form $\texttt{N.i.g.c.n}$, where

• $N$ is the level of $H$
• $i$ is the index of $H$
• $g$ is the genus of $H$
• $c$ is a base-26 ordinal that uniquely identifies the Gassmann class of $H$ among groups of the same level, index, and genus.
• $n$ is a positive integer that distinguishes nonconjugate subgroups of the same level, index, genus, and Gassmann class.

When $\det(H)=\widehat\Z^\times$ and $-I\not \in H$ this label has the form $\texttt{N.i.g-M.c.m.n}$, where $N$, $i$, and $g$ are as above, and

• $N$ is the level of $H$
• $i$ is the index of $H$
• $g$ is the genus of $H$
• $M$ is the level of $H':=\langle H, -I\rangle$
• $c$ is the base-26 ordinal that identifies the Gassmann class of $H'$
• $m$ is the positive integer in the label $\texttt{M.j.g.c.m}$ of $H'$, where $i=2j$.
• $n$ is a positive integer that distinguishes $H$ from nonconjugate refinements of $H'$ of the same level.