Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| 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 | $2^{4}\cdot4^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-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: | 48.768.17.21013 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&26\\0&37\end{bmatrix}$, $\begin{bmatrix}37&20\\0&41\end{bmatrix}$, $\begin{bmatrix}43&0\\12&37\end{bmatrix}$, $\begin{bmatrix}43&14\\36&17\end{bmatrix}$, $\begin{bmatrix}43&38\\12&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.384.17.hv.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{73}\cdot3^{27}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{4}$ |
| Newforms: | 24.2.a.a$^{2}$, 48.2.a.a, 72.2.a.a, 72.2.d.b, 96.2.d.a$^{2}$, 144.2.a.b, 288.2.a.b, 288.2.a.c, 288.2.a.d$^{2}$, 288.2.d.b |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=7,47$, 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.dj.1.9 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.192.1-48.g.1.4 | $48$ | $4$ | $4$ | $1$ | $0$ | $1^{8}\cdot2^{4}$ |
| 48.384.7-48.ci.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.ci.2.23 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-24.dj.1.12 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.gr.2.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.gr.2.29 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.gt.2.13 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.gt.2.20 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.9-48.hr.1.33 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{4}$ |
| 48.384.9-48.hr.1.43 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{4}$ |
| 48.384.9-48.bfr.2.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfr.2.20 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bft.2.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bft.2.29 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
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.dm.3.4 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.dm.4.4 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.dq.3.8 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.dq.4.8 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.ek.1.4 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.ek.2.4 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.eo.3.8 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.eo.4.8 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.41-48.kl.2.21 | $48$ | $2$ | $2$ | $41$ | $4$ | $1^{12}\cdot2^{4}\cdot4$ |
| 48.1536.41-48.ko.1.10 | $48$ | $2$ | $2$ | $41$ | $3$ | $1^{12}\cdot2^{4}\cdot4$ |
| 48.1536.41-48.pu.1.4 | $48$ | $2$ | $2$ | $41$ | $3$ | $1^{12}\cdot2^{4}\cdot4$ |
| 48.1536.41-48.pv.1.4 | $48$ | $2$ | $2$ | $41$ | $4$ | $1^{12}\cdot2^{4}\cdot4$ |
| 48.1536.41-48.bip.1.12 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot4^{4}$ |
| 48.1536.41-48.bip.2.12 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot4^{4}$ |
| 48.1536.41-48.bit.1.12 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot4^{4}$ |
| 48.1536.41-48.bit.2.12 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot4^{4}$ |
| 48.2304.65-48.i.2.8 | $48$ | $3$ | $3$ | $65$ | $3$ | $1^{24}\cdot2^{12}$ |