Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| Index: | $768$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $17 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (none of which are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot6^{8}\cdot12^{4}\cdot16^{4}\cdot48^{4}$ | Cusp orbits | $4^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48CM17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.768.17.21316 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&42\\24&37\end{bmatrix}$, $\begin{bmatrix}19&6\\24&29\end{bmatrix}$, $\begin{bmatrix}23&41\\0&5\end{bmatrix}$, $\begin{bmatrix}47&21\\24&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.384.17.qd.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{82}\cdot3^{27}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{4}$ |
| Newforms: | 24.2.a.a, 72.2.a.a, 72.2.d.b, 96.2.d.a$^{2}$, 144.2.a.b, 192.2.a.b, 192.2.a.d, 288.2.d.b, 576.2.a.c$^{2}$, 576.2.a.g, 576.2.a.h |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=5,7,47,53$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.7-24.eb.1.8 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.192.1-48.bt.1.4 | $48$ | $4$ | $4$ | $1$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.384.7-48.dd.2.16 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.dd.2.27 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-24.eb.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.gr.1.8 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.gr.1.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.gt.2.20 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.gt.2.25 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.9-48.mn.1.11 | $48$ | $2$ | $2$ | $9$ | $2$ | $2^{4}$ |
| 48.384.9-48.mn.1.15 | $48$ | $2$ | $2$ | $9$ | $2$ | $2^{4}$ |
| 48.384.9-48.bfj.1.15 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfj.1.25 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfl.2.12 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfl.2.18 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.1536.33-48.kg.2.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.kg.4.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.kk.3.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.kk.4.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.ms.3.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.ms.4.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.mw.2.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.mw.4.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.2304.65-48.nl.1.9 | $48$ | $3$ | $3$ | $65$ | $6$ | $1^{24}\cdot2^{12}$ |