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: | $\begin{bmatrix}14&23\\13&104\end{bmatrix}$, $\begin{bmatrix}66&51\\53&48\end{bmatrix}$, $\begin{bmatrix}103&4\\6&13\end{bmatrix}$, $\begin{bmatrix}104&25\\51&14\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.48.0.bs.2 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $8$ |
| Cyclic 112-torsion field degree: | $384$ |
| 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 |
|---|---|---|---|---|---|
| 16.48.0-16.g.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.bu.1.11 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 112.48.0-112.e.2.3 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-112.e.2.10 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-16.g.1.14 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.1.9 | $112$ | $2$ | $2$ | $0$ | $?$ |
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.j.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.s.2.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.bk.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.br.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dm.2.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dp.2.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.eb.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.eg.2.1 | $112$ | $2$ | $2$ | $1$ |
| 224.192.1-224.a.1.7 | $224$ | $2$ | $2$ | $1$ |
| 224.192.1-224.m.2.5 | $224$ | $2$ | $2$ | $1$ |
| 224.192.1-224.q.1.5 | $224$ | $2$ | $2$ | $1$ |
| 224.192.1-224.u.2.1 | $224$ | $2$ | $2$ | $1$ |
| 224.192.3-224.z.1.8 | $224$ | $2$ | $2$ | $3$ |
| 224.192.3-224.bd.1.6 | $224$ | $2$ | $2$ | $3$ |
| 224.192.3-224.bt.1.6 | $224$ | $2$ | $2$ | $3$ |
| 224.192.3-224.cf.1.5 | $224$ | $2$ | $2$ | $3$ |