Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $1152$ | $\PSL_2$-index: | $576$ | ||||
| Genus: | $29 = 1 + \frac{ 576 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (none of which are rational) | Cusp widths | $3^{16}\cdot6^{8}\cdot12^{8}\cdot48^{8}$ | Cusp orbits | $2^{6}\cdot4^{7}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $7 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $7 \le \gamma \le 12$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.1152.29.1170 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&15\\24&19\end{bmatrix}$, $\begin{bmatrix}19&18\\36&41\end{bmatrix}$, $\begin{bmatrix}37&9\\0&7\end{bmatrix}$, $\begin{bmatrix}37&12\\24&25\end{bmatrix}$, $\begin{bmatrix}47&39\\36&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.576.29.hh.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{120}\cdot3^{48}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{8}$ |
| Newforms: | 24.2.a.a$^{3}$, 36.2.a.a$^{3}$, 48.2.a.a, 72.2.a.a$^{3}$, 96.2.d.a$^{3}$, 144.2.a.a, 144.2.a.b$^{2}$, 288.2.d.a$^{2}$, 288.2.d.b$^{3}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=7,31,103,127,151,271,439,727$, 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 |
|---|---|---|---|---|---|---|
| 24.576.13-24.ll.2.4 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{8}\cdot2^{4}$ |
| 48.384.7-48.gt.2.20 | $48$ | $3$ | $3$ | $7$ | $0$ | $1^{10}\cdot2^{6}$ |
| 48.576.13-48.bh.1.23 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{8}$ |
| 48.576.13-48.bh.1.45 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{8}$ |
| 48.576.13-48.by.1.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}\cdot2^{4}$ |
| 48.576.13-48.by.1.24 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}\cdot2^{4}$ |
| 48.576.13-24.ll.2.19 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}\cdot2^{4}$ |
| 48.576.15-48.py.2.12 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}$ |
| 48.576.15-48.py.2.24 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}$ |
| 48.576.15-48.qf.1.12 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}$ |
| 48.576.15-48.qf.1.24 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}$ |
| 48.576.15-48.vf.1.13 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}$ |
| 48.576.15-48.vf.1.19 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}$ |
| 48.576.15-48.vi.2.3 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}$ |
| 48.576.15-48.vi.2.11 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.2304.57-48.di.1.4 | $48$ | $2$ | $2$ | $57$ | $0$ | $2^{10}\cdot4^{2}$ |
| 48.2304.57-48.dq.1.4 | $48$ | $2$ | $2$ | $57$ | $0$ | $2^{10}\cdot4^{2}$ |
| 48.2304.57-48.ep.2.6 | $48$ | $2$ | $2$ | $57$ | $0$ | $2^{10}\cdot4^{2}$ |
| 48.2304.57-48.ex.1.7 | $48$ | $2$ | $2$ | $57$ | $0$ | $2^{10}\cdot4^{2}$ |
| 48.2304.65-48.i.2.8 | $48$ | $2$ | $2$ | $65$ | $3$ | $1^{20}\cdot2^{8}$ |
| 48.2304.65-48.hu.2.9 | $48$ | $2$ | $2$ | $65$ | $6$ | $1^{20}\cdot2^{8}$ |
| 48.2304.65-48.lf.1.8 | $48$ | $2$ | $2$ | $65$ | $3$ | $1^{20}\cdot2^{8}$ |
| 48.2304.65-48.nl.1.9 | $48$ | $2$ | $2$ | $65$ | $6$ | $1^{20}\cdot2^{8}$ |
| 48.2304.65-48.yj.2.11 | $48$ | $2$ | $2$ | $65$ | $6$ | $1^{20}\cdot2^{8}$ |
| 48.2304.65-48.zo.1.12 | $48$ | $2$ | $2$ | $65$ | $2$ | $1^{20}\cdot2^{8}$ |
| 48.2304.65-48.bah.2.9 | $48$ | $2$ | $2$ | $65$ | $6$ | $1^{20}\cdot2^{8}$ |
| 48.2304.65-48.bca.1.11 | $48$ | $2$ | $2$ | $65$ | $2$ | $1^{20}\cdot2^{8}$ |
| 48.2304.65-48.bfx.2.7 | $48$ | $2$ | $2$ | $65$ | $0$ | $2^{8}\cdot4^{5}$ |
| 48.2304.65-48.bgf.1.11 | $48$ | $2$ | $2$ | $65$ | $0$ | $2^{8}\cdot4^{5}$ |
| 48.2304.65-48.bhc.2.11 | $48$ | $2$ | $2$ | $65$ | $0$ | $2^{8}\cdot4^{5}$ |
| 48.2304.65-48.bhk.2.13 | $48$ | $2$ | $2$ | $65$ | $0$ | $2^{8}\cdot4^{5}$ |