Invariants
| Level: | $112$ | $\SL_2$-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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}15&104\\99&79\end{bmatrix}$, $\begin{bmatrix}35&24\\17&61\end{bmatrix}$, $\begin{bmatrix}59&104\\20&109\end{bmatrix}$, $\begin{bmatrix}81&0\\103&83\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.bg.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $384$ |
| Full 112-torsion field degree: | $516096$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ x^{2} - 56 y^{2} - 7 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.ba.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 112.48.0-8.ba.1.8 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bf.1.6 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bf.1.7 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.2.1 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.2.14 | $112$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 112.192.1-112.ce.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.cg.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.cm.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.co.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dk.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dm.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.ds.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.du.1.5 | $112$ | $2$ | $2$ | $1$ |