Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-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: | $\begin{bmatrix}17&40\\48&85\end{bmatrix}$, $\begin{bmatrix}55&16\\76&101\end{bmatrix}$, $\begin{bmatrix}57&88\\72&95\end{bmatrix}$, $\begin{bmatrix}87&72\\56&93\end{bmatrix}$, $\begin{bmatrix}95&8\\40&61\end{bmatrix}$, $\begin{bmatrix}111&96\\88&39\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.384.5-112.cz.2.1, 112.384.5-112.cz.2.2, 112.384.5-112.cz.2.3, 112.384.5-112.cz.2.4, 112.384.5-112.cz.2.5, 112.384.5-112.cz.2.6, 112.384.5-112.cz.2.7, 112.384.5-112.cz.2.8, 112.384.5-112.cz.2.9, 112.384.5-112.cz.2.10, 112.384.5-112.cz.2.11, 112.384.5-112.cz.2.12, 112.384.5-112.cz.2.13, 112.384.5-112.cz.2.14, 112.384.5-112.cz.2.15, 112.384.5-112.cz.2.16, 112.384.5-112.cz.2.17, 112.384.5-112.cz.2.18, 112.384.5-112.cz.2.19, 112.384.5-112.cz.2.20 |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $768$ |
| Full 112-torsion field degree: | $258048$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=3,31$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.96.1.g.1 | $8$ | $2$ | $2$ | $1$ | $0$ |
| 112.96.2.d.1 | $112$ | $2$ | $2$ | $2$ | $?$ |
| 112.96.2.h.1 | $112$ | $2$ | $2$ | $2$ | $?$ |
| 112.96.3.bz.2 | $112$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 112.384.13.ed.2 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.ee.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.eg.2 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.eh.2 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.ei.2 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.ej.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.ek.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.el.2 | $112$ | $2$ | $2$ | $13$ |
| 112.384.17.er.1 | $112$ | $2$ | $2$ | $17$ |
| 112.384.17.es.2 | $112$ | $2$ | $2$ | $17$ |
| 112.384.17.et.2 | $112$ | $2$ | $2$ | $17$ |
| 112.384.17.eu.1 | $112$ | $2$ | $2$ | $17$ |