Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $8^{16}\cdot16^{4}$ | Cusp orbits | $2^{4}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 7$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16B7 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}41&0\\4&51\end{bmatrix}$, $\begin{bmatrix}81&8\\20&23\end{bmatrix}$, $\begin{bmatrix}97&40\\80&107\end{bmatrix}$, $\begin{bmatrix}103&44\\40&55\end{bmatrix}$, $\begin{bmatrix}111&32\\40&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.384.7-112.n.1.1, 112.384.7-112.n.1.2, 112.384.7-112.n.1.3, 112.384.7-112.n.1.4, 112.384.7-112.n.1.5, 112.384.7-112.n.1.6, 112.384.7-112.n.1.7, 112.384.7-112.n.1.8, 112.384.7-112.n.1.9, 112.384.7-112.n.1.10, 112.384.7-112.n.1.11, 112.384.7-112.n.1.12, 112.384.7-112.n.1.13, 112.384.7-112.n.1.14, 112.384.7-112.n.1.15, 112.384.7-112.n.1.16 |
| Cyclic 112-isogeny field degree: | $32$ |
| Cyclic 112-torsion field degree: | $768$ |
| Full 112-torsion field degree: | $258048$ |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.96.2.b.1 | $16$ | $2$ | $2$ | $2$ | $0$ |
| 56.96.3.x.1 | $56$ | $2$ | $2$ | $3$ | $1$ |
| 112.96.2.d.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.n.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.n.2 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.r.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.r.2 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.eg.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.eg.2 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.er.1 | $112$ | $2$ | $2$ | $13$ |
| 112.384.13.er.2 | $112$ | $2$ | $2$ | $13$ |