Label of a Shimura curve (awaiting review)

Let $(O,\mu)$ be a polarized order in an indefinite division quaternion algebra $B$ over $\Q$ of discriminant $D$. Let $H \le \Aut_{\pm\mu}(O)\ltimes \widehat{O}^\times$ be an open compact subgroup of the enhanced semidirect product of level $N$. The label of the Shimura Curve $X_H$ has the form $\mathtt{D.d.\delta.N.i.g.c.n}$, where

• $D$ is the discriminant of $B$,
• $d$ is the reduced discriminant of $O$, which is omitted when $O$ is maximal,
• $\delta$ is the degree of the polarized element $\mu$,
• $N$ is the level of $H$,
• $i$ is the index of $H$,
• $g$ is the genus of $X_H$,
• $c$ is a base-26 ordinal that uniquely identifies the Gassman class of $H$ among groups of the same level, index, and genus, and
• $n$ is a positive integer that distinguishes nonconjugate subgroups of the same level, index, genus, and Gassmann class.