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 | $2^{6}\cdot4^{3}$ | ||
| 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: | 48.384.5.24 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&12\\8&13\end{bmatrix}$, $\begin{bmatrix}9&2\\16&19\end{bmatrix}$, $\begin{bmatrix}25&18\\0&23\end{bmatrix}$, $\begin{bmatrix}41&0\\24&13\end{bmatrix}$, $\begin{bmatrix}41&14\\16&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.5.l.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{33}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1\cdot2^{2}$ |
| Newforms: | 32.2.a.a, 1152.2.k.b$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y t + z w $ |
| $=$ | $y^{2} + 2 y t - z^{2} - 2 z w + w^{2} - t^{2}$ | |
| $=$ | $6 x^{2} - y t + z^{2} + z w + t^{2}$ |
Rational points
This modular curve has no real points, 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 |
|---|---|---|---|---|---|---|
| 8.192.1-8.g.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 48.192.1-8.g.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 48.192.2-48.b.1.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.2-48.b.1.18 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.2-48.f.2.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.2-48.f.2.32 | $48$ | $2$ | $2$ | $2$ | $0$ | $1\cdot2$ |
| 48.192.3-48.bf.1.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.bf.1.31 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.13-48.cf.1.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ch.1.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ci.1.3 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.cj.1.2 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ck.2.3 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.cl.2.2 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.cm.2.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.cn.2.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bo.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.bp.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.bq.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.br.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{4}$ |
| 48.1152.37-48.rb.1.2 | $48$ | $3$ | $3$ | $37$ | $1$ | $1^{8}\cdot2^{4}\cdot8^{2}$ |
| 48.1536.41-48.fu.1.6 | $48$ | $4$ | $4$ | $41$ | $0$ | $1^{8}\cdot2^{6}\cdot8^{2}$ |