Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| Index: | $384$ | $\PSL_2$-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: | 48.384.17.1746 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&36\\24&41\end{bmatrix}$, $\begin{bmatrix}19&5\\0&31\end{bmatrix}$, $\begin{bmatrix}19&46\\24&29\end{bmatrix}$, $\begin{bmatrix}29&1\\24&1\end{bmatrix}$, $\begin{bmatrix}41&11\\0&35\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 48.768.17-48.qd.2.1, 48.768.17-48.qd.2.2, 48.768.17-48.qd.2.3, 48.768.17-48.qd.2.4, 48.768.17-48.qd.2.5, 48.768.17-48.qd.2.6, 48.768.17-48.qd.2.7, 48.768.17-48.qd.2.8, 48.768.17-48.qd.2.9, 48.768.17-48.qd.2.10, 48.768.17-48.qd.2.11, 48.768.17-48.qd.2.12, 48.768.17-48.qd.2.13, 48.768.17-48.qd.2.14, 48.768.17-48.qd.2.15, 48.768.17-48.qd.2.16 |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $3072$ |
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 |
|---|---|---|---|---|---|---|
| 24.192.7.eb.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.96.1.bt.1 | $48$ | $4$ | $4$ | $1$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.192.7.dd.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.192.7.gr.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.192.7.gt.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.192.9.mn.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $2^{4}$ |
| 48.192.9.bfj.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.192.9.bfl.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.33.kg.2 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.768.33.kg.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.768.33.kk.3 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.768.33.kk.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.768.33.ms.3 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.768.33.ms.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.768.33.mw.2 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.768.33.mw.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1152.65.nl.1 | $48$ | $3$ | $3$ | $65$ | $6$ | $1^{24}\cdot2^{12}$ |