Invariants
| Level: | $112$ | $\SL_2$-level: | $112$ | Newform level: | $224$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot7^{2}\cdot8^{2}\cdot14\cdot28\cdot56^{2}$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 11$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 11$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56O11 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}4&89\\101&48\end{bmatrix}$, $\begin{bmatrix}14&69\\29&54\end{bmatrix}$, $\begin{bmatrix}21&108\\40&89\end{bmatrix}$, $\begin{bmatrix}27&32\\44&71\end{bmatrix}$, $\begin{bmatrix}34&109\\91&108\end{bmatrix}$, $\begin{bmatrix}94&77\\69&102\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.192.11.ff.2 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $2$ |
| Cyclic 112-torsion field degree: | $48$ |
| Full 112-torsion field degree: | $129024$ |
Models
Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations
| $ 0 $ | $=$ | $ x z - x w - x u - x v + x b + y z + u s $ |
| $=$ | $x w - x u + x b + y z - y w - y u - t b + u r + v s$ | |
| $=$ | $2 x z - x w + y u - w t + t v - v s$ | |
| $=$ | $x z + x w + z s - t b + u r - u s$ | |
| $=$ | $\cdots$ |
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 |
|---|---|---|---|---|---|
| 112.192.5-56.bl.1.1 | $112$ | $2$ | $2$ | $5$ | $?$ |
| 112.192.5-56.bl.1.31 | $112$ | $2$ | $2$ | $5$ | $?$ |