Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| Index: | $768$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $13 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (none of which are rational) | Cusp widths | $1^{8}\cdot2^{4}\cdot3^{8}\cdot4^{4}\cdot6^{4}\cdot12^{4}\cdot16^{4}\cdot48^{4}$ | Cusp orbits | $2^{4}\cdot4^{4}\cdot8^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AH13 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.768.13.11000 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}23&9\\36&23\end{bmatrix}$, $\begin{bmatrix}35&14\\36&41\end{bmatrix}$, $\begin{bmatrix}41&25\\24&35\end{bmatrix}$, $\begin{bmatrix}43&27\\36&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.384.13.lt.1 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: | $2^{54}\cdot3^{17}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{5}$ |
| Newforms: | 24.2.a.a, 24.2.f.a$^{2}$, 72.2.a.a, 96.2.d.a, 144.2.a.b, 192.2.c.a, 288.2.d.b |
Rational points
This modular curve has no $\Q_p$ points for $p=7,37$, 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 |
|---|---|---|---|---|---|---|
| 24.384.5-24.fz.4.10 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.384.5-24.fz.4.13 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.384.5-48.kc.1.18 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.384.5-48.kc.1.19 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.384.5-48.kr.3.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{4}$ |
| 48.384.5-48.kr.3.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{4}$ |
| 48.384.7-48.et.1.18 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.384.7-48.et.1.25 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.384.7-48.fj.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.384.7-48.fj.2.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.384.7-48.gl.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
| 48.384.7-48.gl.1.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
| 48.384.7-48.gt.2.14 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
| 48.384.7-48.gt.2.20 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
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.dq.4.8 | $48$ | $2$ | $2$ | $33$ | $1$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.hi.4.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.jo.4.5 | $48$ | $2$ | $2$ | $33$ | $1$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.kk.4.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.ou.3.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.pa.4.8 | $48$ | $2$ | $2$ | $33$ | $0$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.qq.3.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.ra.3.7 | $48$ | $2$ | $2$ | $33$ | $0$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.2304.57-48.ep.2.6 | $48$ | $3$ | $3$ | $57$ | $0$ | $1^{10}\cdot2^{13}\cdot4^{2}$ |