Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot16^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16J3 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}15&2\\24&61\end{bmatrix}$, $\begin{bmatrix}23&50\\56&101\end{bmatrix}$, $\begin{bmatrix}71&28\\40&81\end{bmatrix}$, $\begin{bmatrix}71&88\\64&77\end{bmatrix}$, $\begin{bmatrix}81&76\\64&93\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.96.3.ch.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $384$ |
| 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 |
|---|---|---|---|---|---|
| 8.96.0-8.l.2.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 112.96.0-8.l.2.8 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.96.1-112.a.1.4 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.96.1-112.a.1.23 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 112.96.2-112.d.2.11 | $112$ | $2$ | $2$ | $2$ | $?$ |
| 112.96.2-112.d.2.24 | $112$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 112.384.5-112.bp.3.11 | $112$ | $2$ | $2$ | $5$ |
| 112.384.5-112.ca.2.15 | $112$ | $2$ | $2$ | $5$ |
| 112.384.5-112.eg.2.15 | $112$ | $2$ | $2$ | $5$ |
| 112.384.5-112.ek.2.6 | $112$ | $2$ | $2$ | $5$ |
| 112.384.7-112.bk.1.10 | $112$ | $2$ | $2$ | $7$ |
| 112.384.7-112.bl.1.10 | $112$ | $2$ | $2$ | $7$ |
| 112.384.7-112.bo.2.10 | $112$ | $2$ | $2$ | $7$ |
| 112.384.7-112.bp.1.14 | $112$ | $2$ | $2$ | $7$ |
| 112.384.7-112.bs.1.14 | $112$ | $2$ | $2$ | $7$ |
| 112.384.7-112.bt.1.14 | $112$ | $2$ | $2$ | $7$ |
| 112.384.7-112.bw.1.14 | $112$ | $2$ | $2$ | $7$ |
| 112.384.7-112.bx.1.14 | $112$ | $2$ | $2$ | $7$ |