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 | $2^{8}\cdot4^{4}$ | ||
| 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: | 48.768.17.22620 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}17&20\\0&29\end{bmatrix}$, $\begin{bmatrix}23&3\\36&7\end{bmatrix}$, $\begin{bmatrix}35&33\\24&17\end{bmatrix}$, $\begin{bmatrix}41&36\\0&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.384.17.bmm.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $8$ |
| Full 48-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{74}\cdot3^{27}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{4}$ |
| Newforms: | 24.2.a.a, 24.2.d.a$^{2}$, 72.2.a.a, 72.2.d.b, 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 $\Q_p$ points for $p=5,53$, 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.eq.1.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.192.1-48.ek.1.7 | $48$ | $4$ | $4$ | $1$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.384.7-48.ej.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.ej.2.29 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-24.eq.1.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.hw.1.8 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.hw.1.13 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.hx.2.8 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.hx.2.18 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.9-48.bfh.2.14 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfh.2.18 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfk.1.20 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfk.1.25 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bgp.1.4 | $48$ | $2$ | $2$ | $9$ | $2$ | $2^{4}$ |
| 48.384.9-48.bgp.1.15 | $48$ | $2$ | $2$ | $9$ | $2$ | $2^{4}$ |
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.vu.2.6 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.vu.3.6 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.vv.2.6 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.vv.3.7 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.xa.1.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.xa.3.6 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.xb.1.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.1536.33-48.xb.3.7 | $48$ | $2$ | $2$ | $33$ | $2$ | $2^{6}\cdot4$ |
| 48.2304.65-48.bsb.2.8 | $48$ | $3$ | $3$ | $65$ | $6$ | $1^{24}\cdot2^{12}$ |