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: | $10$ (of which $2$ are rational) | Cusp widths | $2^{8}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}5&32\\82&7\end{bmatrix}$, $\begin{bmatrix}45&96\\68&23\end{bmatrix}$, $\begin{bmatrix}53&16\\108&29\end{bmatrix}$, $\begin{bmatrix}97&24\\86&93\end{bmatrix}$, $\begin{bmatrix}105&48\\22&77\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.48.0.d.2 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $768$ |
| Full 112-torsion field degree: | $516096$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 112.48.0-8.i.1.2 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-112.e.2.3 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-112.e.2.30 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-112.f.1.11 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-112.f.1.22 | $112$ | $2$ | $2$ | $0$ | $?$ |
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.2.4 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.b.2.4 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.c.1.4 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.d.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.e.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.f.2.8 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.g.2.4 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.h.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.i.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.j.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.k.2.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.l.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.192.3-112.bm.2.4 | $112$ | $2$ | $2$ | $3$ |
| 112.192.3-112.bo.2.6 | $112$ | $2$ | $2$ | $3$ |
| 112.192.3-112.bt.2.2 | $112$ | $2$ | $2$ | $3$ |
| 112.192.3-112.bw.2.2 | $112$ | $2$ | $2$ | $3$ |