Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}1&64\\76&45\end{bmatrix}$, $\begin{bmatrix}21&48\\94&1\end{bmatrix}$, $\begin{bmatrix}29&40\\32&3\end{bmatrix}$, $\begin{bmatrix}55&48\\20&49\end{bmatrix}$, $\begin{bmatrix}83&104\\26&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.z.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $384$ |
| Full 112-torsion field degree: | $516096$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 7 x^{2} - 8 y^{2} + 7 z^{2} $ |
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 |
|---|---|---|---|---|---|
| 16.48.0-8.i.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 112.48.0-8.i.1.2 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.1.1 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.1.8 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.1.9 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.1.16 | $112$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 112.192.1-112.a.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.a.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-56.f.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.j.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.j.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.m.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.m.2.7 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.p.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.p.2.6 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-56.bq.1.4 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-56.cb.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-56.cf.1.4 | $112$ | $2$ | $2$ | $1$ |
| 112.192.3-112.bn.1.6 | $112$ | $2$ | $2$ | $3$ |
| 112.192.3-112.bs.2.7 | $112$ | $2$ | $2$ | $3$ |
| 112.192.3-112.ca.2.6 | $112$ | $2$ | $2$ | $3$ |
| 112.192.3-112.cm.2.7 | $112$ | $2$ | $2$ | $3$ |