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 | $2^{3}\cdot4$ | ||
| 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: | 48.96.0.1639 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}17&9\\0&43\end{bmatrix}$, $\begin{bmatrix}21&28\\40&13\end{bmatrix}$, $\begin{bmatrix}27&34\\28&9\end{bmatrix}$, $\begin{bmatrix}39&32\\40&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bi.1 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 $ | $=$ | $ 12 x^{2} + 3 y^{2} - 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 |
|---|---|---|---|---|---|
| 16.48.0-16.f.1.11 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.by.2.9 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.f.1.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.g.1.17 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.g.1.31 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.p.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.x.1.7 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bq.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bv.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cm.1.8 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cp.2.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.db.2.6 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dg.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.ia.2.1 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.gv.1.23 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.wr.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wz.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xx.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yf.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zd.1.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zl.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.baj.2.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bar.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.fu.2.18 | $240$ | $5$ | $5$ | $16$ |