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 | $2^{4}\cdot4^{2}\cdot8$ | ||
| 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: | $\begin{bmatrix}9&40\\36&103\end{bmatrix}$, $\begin{bmatrix}67&96\\38&99\end{bmatrix}$, $\begin{bmatrix}69&72\\64&71\end{bmatrix}$, $\begin{bmatrix}99&88\\12&105\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.384.5-112.ch.2.1, 112.384.5-112.ch.2.2, 112.384.5-112.ch.2.3, 112.384.5-112.ch.2.4, 112.384.5-112.ch.2.5, 112.384.5-112.ch.2.6, 112.384.5-112.ch.2.7, 112.384.5-112.ch.2.8, 224.384.5-112.ch.2.1, 224.384.5-112.ch.2.2, 224.384.5-112.ch.2.3, 224.384.5-112.ch.2.4 |
| Cyclic 112-isogeny field degree: | $8$ |
| Cyclic 112-torsion field degree: | $384$ |
| Full 112-torsion field degree: | $258048$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.96.3.z.1 | $16$ | $2$ | $2$ | $3$ | $0$ |
| 56.96.1.cf.1 | $56$ | $2$ | $2$ | $1$ | $1$ |
| 112.96.1.h.2 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.96.1.j.1 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.96.3.bs.2 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.96.3.bt.2 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.96.3.cp.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 |
|---|---|---|---|---|
| 224.384.17.bv.3 | $224$ | $2$ | $2$ | $17$ |
| 224.384.17.bv.4 | $224$ | $2$ | $2$ | $17$ |
| 224.384.17.bw.3 | $224$ | $2$ | $2$ | $17$ |
| 224.384.17.bw.4 | $224$ | $2$ | $2$ | $17$ |