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 | $1^{2}\cdot2^{3}\cdot16$ | 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: | 16D0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}6&21\\11&24\end{bmatrix}$, $\begin{bmatrix}13&24\\42&11\end{bmatrix}$, $\begin{bmatrix}14&31\\85&72\end{bmatrix}$, $\begin{bmatrix}32&45\\107&50\end{bmatrix}$, $\begin{bmatrix}32&111\\85&2\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.24.0.e.2 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.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
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.d.2.10 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.e.1.9 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.j.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.l.2.9 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.v.2.10 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.w.1.9 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.y.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bb.2.9 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.be.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bf.2.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bm.2.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bn.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bs.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bt.2.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bw.2.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bx.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.bg.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bh.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bk.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bl.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bq.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.br.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.by.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bz.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.384.11-112.q.2.11 | $112$ | $8$ | $8$ | $11$ |