Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | Newform level: | $1$ | ||
| 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: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16O5 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}61&40\\36&17\end{bmatrix}$, $\begin{bmatrix}75&64\\8&57\end{bmatrix}$, $\begin{bmatrix}97&72\\28&33\end{bmatrix}$, $\begin{bmatrix}99&64\\52&1\end{bmatrix}$, $\begin{bmatrix}101&48\\8&105\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.192.5.l.2 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $192$ |
| Full 112-torsion field degree: | $129024$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=31$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.192.1-8.g.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ |
| 112.192.1-8.g.2.1 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.192.2-112.b.1.1 | $112$ | $2$ | $2$ | $2$ | $?$ |
| 112.192.2-112.b.1.19 | $112$ | $2$ | $2$ | $2$ | $?$ |
| 112.192.2-112.f.2.1 | $112$ | $2$ | $2$ | $2$ | $?$ |
| 112.192.2-112.f.2.28 | $112$ | $2$ | $2$ | $2$ | $?$ |
| 112.192.3-112.bf.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.192.3-112.bf.1.31 | $112$ | $2$ | $2$ | $3$ | $?$ |