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: | $\begin{bmatrix}4&37\\93&60\end{bmatrix}$, $\begin{bmatrix}88&23\\77&58\end{bmatrix}$, $\begin{bmatrix}95&94\\102&79\end{bmatrix}$, $\begin{bmatrix}102&63\\47&6\end{bmatrix}$, $\begin{bmatrix}106&61\\35&36\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.24.0.f.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $768$ |
| 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 |
|---|---|---|---|---|---|
| 8.24.0-8.n.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 112.24.0-8.n.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 |
|---|---|---|---|---|
| 112.96.0-112.d.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.f.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.k.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.l.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.u.2.3 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.x.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.z.2.3 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.ba.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bg.1.11 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bh.2.13 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bo.2.9 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bp.1.9 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bu.1.12 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bv.2.15 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.by.2.13 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bz.1.11 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.bi.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bj.2.3 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bm.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bn.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bs.2.4 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bt.2.6 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.ca.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.cb.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.384.11-112.r.2.59 | $112$ | $8$ | $8$ | $11$ |