Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{8}\cdot16^{2}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.659 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}17&18\\24&25\end{bmatrix}$, $\begin{bmatrix}21&1\\40&39\end{bmatrix}$, $\begin{bmatrix}23&23\\24&17\end{bmatrix}$, $\begin{bmatrix}31&3\\0&5\end{bmatrix}$, $\begin{bmatrix}41&11\\8&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.k.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $12288$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 3 x^{2} - 2 y^{2} + 12 z^{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 |
|---|---|---|---|---|---|
| 16.48.0-8.q.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-8.q.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.16 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.17 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.1.32 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.