Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $1152$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{12}\cdot16^{4}$ | Cusp orbits | $2^{4}\cdot4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16O5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.5.2910 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&2\\16&15\end{bmatrix}$, $\begin{bmatrix}19&10\\44&37\end{bmatrix}$, $\begin{bmatrix}25&46\\8&39\end{bmatrix}$, $\begin{bmatrix}43&4\\28&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.5.g.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $16$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{33}\cdot3^{6}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2^{2}$ |
| Newforms: | 128.2.e.b, 288.2.a.d, 1152.2.k.b |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x^{2} + y^{2} - y t + z^{2} + z w - w^{2} - t^{2} $ |
| $=$ | $2 y z - y t - z w + 2 w t$ | |
| $=$ | $2 x^{2} + 2 y w + y t - z w + 2 z t - w^{2} - t^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.192.2-16.b.1.12 | $16$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 24.192.1-24.x.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 48.192.1-24.x.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 48.192.2-16.b.1.4 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.2-48.b.1.9 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.2-48.b.1.18 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.13-48.l.2.2 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.n.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.o.1.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.q.2.7 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ce.1.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ci.2.7 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.cr.1.6 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ct.2.6 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.1152.37-48.qk.1.11 | $48$ | $3$ | $3$ | $37$ | $1$ | $1^{8}\cdot2^{4}\cdot8^{2}$ |
| 48.1536.41-48.en.1.22 | $48$ | $4$ | $4$ | $41$ | $1$ | $1^{8}\cdot2^{6}\cdot8^{2}$ |