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: | $24$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{12}\cdot16^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16O5 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}1&0\\62&27\end{bmatrix}$, $\begin{bmatrix}15&24\\42&43\end{bmatrix}$, $\begin{bmatrix}15&48\\90&17\end{bmatrix}$, $\begin{bmatrix}63&72\\90&1\end{bmatrix}$, $\begin{bmatrix}97&56\\88&85\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.384.5-112.dy.1.1, 112.384.5-112.dy.1.2, 112.384.5-112.dy.1.3, 112.384.5-112.dy.1.4, 112.384.5-112.dy.1.5, 112.384.5-112.dy.1.6, 112.384.5-112.dy.1.7, 112.384.5-112.dy.1.8, 112.384.5-112.dy.1.9, 112.384.5-112.dy.1.10, 112.384.5-112.dy.1.11, 112.384.5-112.dy.1.12, 112.384.5-112.dy.1.13, 112.384.5-112.dy.1.14, 112.384.5-112.dy.1.15, 112.384.5-112.dy.1.16 |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $768$ |
| Full 112-torsion field degree: | $258048$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.96.2.h.1 | $16$ | $2$ | $2$ | $2$ | $0$ |
| 56.96.1.ch.1 | $56$ | $2$ | $2$ | $1$ | $1$ |
| 112.96.2.g.1 | $112$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 112.384.13.ea.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.ez.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.fx.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.gd.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.gz.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.hc.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.hf.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.hl.1 | $112$ | $2$ | $2$ | $13$ |