Invariants
| Level: | $112$ | $\SL_2$-level: | $112$ | Newform level: | $1$ | ||
| 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 $6$ are rational) | Cusp widths | $1^{4}\cdot4\cdot7^{4}\cdot16\cdot28\cdot112$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 11$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 11$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 112G11 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}11&4\\98&29\end{bmatrix}$, $\begin{bmatrix}22&87\\45&64\end{bmatrix}$, $\begin{bmatrix}38&29\\99&24\end{bmatrix}$, $\begin{bmatrix}44&55\\75&24\end{bmatrix}$, $\begin{bmatrix}77&60\\16&9\end{bmatrix}$, $\begin{bmatrix}77&64\\40&101\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.192.11.k.2 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $2$ |
| Cyclic 112-torsion field degree: | $24$ |
| Full 112-torsion field degree: | $129024$ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 7.16.0-7.a.1.2 | $7$ | $24$ | $24$ | $0$ | $0$ |
| 16.24.0-8.n.1.8 | $16$ | $16$ | $16$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.192.5-56.bl.1.25 | $56$ | $2$ | $2$ | $5$ | $0$ |
| 112.192.5-56.bl.1.31 | $112$ | $2$ | $2$ | $5$ | $?$ |