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.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by Andrew Sutherland on 2023-01-03 11:00:35

**Referred to by:**

- columns.ec_curvedata.modell_images
- columns.gps_gl2zhat_test.CPlabel
- columns.gps_gl2zhat_test.RZBlabel
- columns.gps_gl2zhat_test.SZlabel
- columns.gps_gl2zhat_test.Slabel
- lmfdb/modular_curves/main.py (lines 325-329)
- lmfdb/modular_curves/main.py (line 936)
- lmfdb/modular_curves/main.py (line 986)
- lmfdb/modular_curves/templates/modcurve.html (lines 64-68)
- lmfdb/modular_curves/templates/modcurve.html (line 74)
- lmfdb/modular_curves/templates/modcurve.html (line 80)
- lmfdb/modular_curves/templates/modcurve.html (line 86)
- lmfdb/modular_curves/templates/modcurve.html (line 92)
- lmfdb/modular_curves/templates/modcurve_isoclass.html (lines 23-32)

**History:**(expand/hide all)

- 2023-01-03 11:00:35 by Andrew Sutherland
- 2023-01-03 11:00:05 by Andrew Sutherland
- 2022-12-12 06:54:57 by Andrew Sutherland
- 2022-12-12 06:53:29 by Andrew Sutherland
- 2022-12-01 15:35:34 by Andrew Sutherland
- 2022-12-01 15:35:08 by Andrew Sutherland
- 2022-12-01 15:34:29 by Andrew Sutherland
- 2022-12-01 15:33:56 by Andrew Sutherland
- 2022-12-01 15:32:54 by Andrew Sutherland
- 2022-11-06 19:38:51 by Andrew Sutherland
- 2022-11-05 17:23:52 by John Voight
- 2022-03-20 18:47:52 by Andrew Sutherland
- 2022-03-20 18:39:01 by Andrew Sutherland

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