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 | $4^{16}\cdot16^{8}$ | Cusp orbits | $2^{2}\cdot4^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16M5 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}3&36\\56&37\end{bmatrix}$, $\begin{bmatrix}15&96\\40&103\end{bmatrix}$, $\begin{bmatrix}47&80\\102&91\end{bmatrix}$, $\begin{bmatrix}89&68\\66&79\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.384.5-112.el.1.1, 112.384.5-112.el.1.2, 112.384.5-112.el.1.3, 112.384.5-112.el.1.4, 112.384.5-112.el.1.5, 112.384.5-112.el.1.6, 112.384.5-112.el.1.7, 112.384.5-112.el.1.8, 224.384.5-112.el.1.1, 224.384.5-112.el.1.2, 224.384.5-112.el.1.3, 224.384.5-112.el.1.4 |
| Cyclic 112-isogeny field degree: | $8$ |
| Cyclic 112-torsion field degree: | $384$ |
| Full 112-torsion field degree: | $258048$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.96.1.d.1 | $16$ | $2$ | $2$ | $1$ | $0$ |
| 56.96.1.ce.1 | $56$ | $2$ | $2$ | $1$ | $1$ |
| 112.96.1.q.1 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.96.3.cj.1 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.96.3.cn.2 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.96.3.cp.2 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.96.3.cv.1 | $112$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 224.384.17.bu.1 | $224$ | $2$ | $2$ | $17$ |
| 224.384.17.bu.2 | $224$ | $2$ | $2$ | $17$ |
| 224.384.17.du.1 | $224$ | $2$ | $2$ | $17$ |
| 224.384.17.du.2 | $224$ | $2$ | $2$ | $17$ |