Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $192$ | ||
| Index: | $768$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $17 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (of which $8$ are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot6^{8}\cdot12^{4}\cdot16^{4}\cdot48^{4}$ | Cusp orbits | $1^{8}\cdot2^{4}\cdot4^{4}$ | ||
| 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: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48CM17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.768.17.116 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&44\\24&11\end{bmatrix}$, $\begin{bmatrix}17&10\\24&5\end{bmatrix}$, $\begin{bmatrix}25&2\\24&5\end{bmatrix}$, $\begin{bmatrix}41&18\\0&7\end{bmatrix}$, $\begin{bmatrix}41&28\\0&17\end{bmatrix}$, $\begin{bmatrix}41&38\\24&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.384.17.ih.1 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: | $2^{74}\cdot3^{15}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{4}$ |
| Newforms: | 24.2.a.a$^{2}$, 24.2.d.a$^{3}$, 48.2.a.a, 64.2.a.a$^{2}$, 96.2.d.a, 192.2.a.a, 192.2.a.b, 192.2.a.c, 192.2.a.d |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $192$ | $96$ | $0$ | $0$ | full Jacobian |
| 16.192.1-16.c.2.2 | $16$ | $4$ | $4$ | $1$ | $0$ | $1^{8}\cdot2^{4}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.192.1-16.c.2.2 | $16$ | $4$ | $4$ | $1$ | $0$ | $1^{8}\cdot2^{4}$ |
| 24.384.7-24.dp.1.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.ck.2.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.ck.2.46 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-24.dp.1.20 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.hw.1.8 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.hw.1.25 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.ic.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.7-48.ic.1.29 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.384.9-48.hn.1.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{4}$ |
| 48.384.9-48.hn.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{4}$ |
| 48.384.9-48.bas.1.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bas.1.29 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bay.1.8 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bay.1.25 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
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.eu.2.2 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.eu.4.2 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.ev.2.3 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.ev.4.5 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.fs.2.2 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.fs.4.2 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.ft.1.3 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.33-48.ft.3.5 | $48$ | $2$ | $2$ | $33$ | $1$ | $2^{6}\cdot4$ |
| 48.1536.41-48.ne.2.3 | $48$ | $2$ | $2$ | $41$ | $1$ | $1^{12}\cdot2^{4}\cdot4$ |
| 48.1536.41-48.nf.1.8 | $48$ | $2$ | $2$ | $41$ | $3$ | $1^{12}\cdot2^{4}\cdot4$ |
| 48.1536.41-48.bcs.1.2 | $48$ | $2$ | $2$ | $41$ | $3$ | $1^{12}\cdot2^{4}\cdot4$ |
| 48.1536.41-48.bct.2.7 | $48$ | $2$ | $2$ | $41$ | $7$ | $1^{12}\cdot2^{4}\cdot4$ |
| 48.1536.41-48.bgv.1.8 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot8^{2}$ |
| 48.1536.41-48.bgy.1.14 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot8^{2}$ |
| 48.1536.41-48.bhb.1.7 | $48$ | $2$ | $2$ | $41$ | $1$ | $12^{2}$ |
| 48.1536.41-48.bhb.3.7 | $48$ | $2$ | $2$ | $41$ | $1$ | $12^{2}$ |
| 48.1536.41-48.bhe.1.4 | $48$ | $2$ | $2$ | $41$ | $1$ | $12^{2}$ |
| 48.1536.41-48.bhe.3.4 | $48$ | $2$ | $2$ | $41$ | $1$ | $12^{2}$ |
| 48.1536.41-48.bhf.1.4 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot8^{2}$ |
| 48.1536.41-48.bhg.1.6 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot8^{2}$ |
| 48.1536.41-48.bhh.1.10 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot8^{2}$ |
| 48.1536.41-48.bhi.1.6 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot8^{2}$ |
| 48.1536.41-48.bhj.1.4 | $48$ | $2$ | $2$ | $41$ | $1$ | $12^{2}$ |
| 48.1536.41-48.bhj.3.4 | $48$ | $2$ | $2$ | $41$ | $1$ | $12^{2}$ |
| 48.1536.41-48.bhq.1.4 | $48$ | $2$ | $2$ | $41$ | $1$ | $12^{2}$ |
| 48.1536.41-48.bhq.3.4 | $48$ | $2$ | $2$ | $41$ | $1$ | $12^{2}$ |
| 48.1536.41-48.bht.1.6 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot8^{2}$ |
| 48.1536.41-48.bhw.1.4 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot8^{2}$ |
| 48.1536.41-48.bjp.1.2 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot4^{4}$ |
| 48.1536.41-48.bjp.3.2 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot4^{4}$ |
| 48.1536.41-48.bjq.1.3 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot4^{4}$ |
| 48.1536.41-48.bjq.3.3 | $48$ | $2$ | $2$ | $41$ | $1$ | $2^{4}\cdot4^{4}$ |
| 48.2304.65-48.dn.2.7 | $48$ | $3$ | $3$ | $65$ | $5$ | $1^{24}\cdot2^{12}$ |