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Modular curves are identified in the literature using several different labeling conventions. In addition to the standard names for the classical modular curves $X(N), X_0(N), X_1(N), X_{\mathrm ns}^+(N), X_{\mathrm sp}^+(N)$, etc., there are a number of labeling systems that are used in different contexts, including:

  • (CP) Cummins and Pauli labels $\texttt{NA}^\texttt{g}$,are used to identify congruence subgroups of $\PSL_2(\Z)$ of level $N$ and genus $g\le 24$ on this site, which is based on [MR:2016709].
    Note that Cummins and Pauli work up to $\PGL_2(\Z)$-conjugacy, thus for some open $H$ in $\GL_2(\widehat\Z)$ the group $\pm H\cap \SL_2(\widehat \Z)$ may have more than one CP label.
  • (RZB) Rouse and Zureick-Brown labels $X\texttt{m}$ with $1\le m\le 727$ are used for modular curves $X_H$ of $2$-power level on this site, based on [10.1007/s40993-015-0013-7].
    Note that Rouse and Zureick-Brown put the action on the right, rather than the left (so one should transpose $H$ when comparing with their tables).
  • (RSZB) Rouse, Sutherland, and Zureick-Brown labels $\texttt{N.i.g.n}$ identify open subgroups of $\GL_2(\widehat\Z)$ according to their level $N$, index $i$, and genus $g$, as defined in [10.1017/fms.2022.38].
  • (S) Sutherland labels $\texttt{NS.a.b.c}$ for groups of prime level $N$ were introduced in [arXiv:1504.07618, MR:3482279] and are used to identify mod-$\ell$ Galois images.
  • (SZ) Sutherland and Zywina labels $\texttt{MA}^\texttt{g}\texttt{-Na}$ are used in [arXiv:1605.03988, MR:3671434] to identify modular curves of prime power level with infinitely many rational points.
Knowl status:
  • Review status: beta
  • Last edited by Andrew Sutherland on 2023-01-03 11:00:35
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