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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.148 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&1\\4&7\end{bmatrix}$, $\begin{bmatrix}9&19\\32&11\end{bmatrix}$, $\begin{bmatrix}11&39\\20&11\end{bmatrix}$, $\begin{bmatrix}33&22\\8&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bp.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $12288$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 6 x^{2} - 3 y^{2} + 4 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 |
|---|---|---|---|---|---|
| 8.48.0-8.bb.2.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.2.29 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.h.1.31 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.bb.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 48.192.1-48.f.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ba.2.7 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bm.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.by.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cg.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cu.2.8 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cy.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.di.1.11 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.ip.2.5 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.hc.2.15 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.xg.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xo.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ym.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yu.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zs.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.baa.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bay.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbg.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.gj.2.20 | $240$ | $5$ | $5$ | $16$ |