Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot4\cdot16$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 16C0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}7&22\\38&39\end{bmatrix}$, $\begin{bmatrix}56&97\\23&18\end{bmatrix}$, $\begin{bmatrix}57&94\\104&7\end{bmatrix}$, $\begin{bmatrix}97&34\\18&41\end{bmatrix}$, $\begin{bmatrix}99&88\\78&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.24.0.g.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $384$ |
| Full 112-torsion field degree: | $1032192$ |
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 |
|---|---|---|---|---|---|
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-8.n.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 112.96.0-112.bc.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bc.2.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bd.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bd.2.3 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.be.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.be.2.3 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bf.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bf.2.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bg.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bg.2.3 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bh.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bh.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bi.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bi.2.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bj.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bj.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.b.2.10 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.d.1.10 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.h.1.10 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.i.1.6 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.r.1.10 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.s.1.10 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.v.1.6 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.w.1.10 | $112$ | $2$ | $2$ | $1$ |
| 112.384.11-112.u.1.13 | $112$ | $8$ | $8$ | $11$ |