Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-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: | $\begin{bmatrix}33&32\\14&85\end{bmatrix}$, $\begin{bmatrix}41&48\\89&11\end{bmatrix}$, $\begin{bmatrix}49&32\\15&111\end{bmatrix}$, $\begin{bmatrix}85&40\\42&65\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.192.5.gn.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $192$ |
| Full 112-torsion field degree: | $129024$ |
Rational points
This modular curve has no $\Q_p$ points for $p=29$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.192.3-16.cl.1.2 | $16$ | $2$ | $2$ | $3$ | $0$ |
| 56.192.1-56.ck.1.5 | $56$ | $2$ | $2$ | $1$ | $0$ |
| 112.192.1-112.bf.1.3 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.192.1-112.bf.1.11 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.192.1-112.bf.2.2 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.192.1-112.bf.2.12 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.192.1-56.ck.1.6 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.192.3-16.cl.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.192.3-112.fq.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.192.3-112.fq.1.6 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.192.3-112.fq.2.6 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.192.3-112.fq.2.10 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.192.3-112.fr.1.2 | $112$ | $2$ | $2$ | $3$ | $?$ |
| 112.192.3-112.fr.1.5 | $112$ | $2$ | $2$ | $3$ | $?$ |