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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | 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: | 16H0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}4&53\\85&68\end{bmatrix}$, $\begin{bmatrix}24&81\\23&18\end{bmatrix}$, $\begin{bmatrix}42&29\\39&104\end{bmatrix}$, $\begin{bmatrix}92&69\\41&72\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.48.0.bf.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $192$ |
| 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 |
|---|---|---|---|---|---|
| 16.48.0-8.bb.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-8.bb.1.7 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 112.48.0-112.e.1.2 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-112.e.1.7 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-112.g.1.1 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-112.g.1.4 | $112$ | $2$ | $2$ | $0$ | $?$ |
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.i.2.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.v.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.bj.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.bu.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.cj.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.cp.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.db.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dd.2.1 | $112$ | $2$ | $2$ | $1$ |