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: | $\begin{bmatrix}4&21\\27&30\end{bmatrix}$, $\begin{bmatrix}52&105\\95&46\end{bmatrix}$, $\begin{bmatrix}101&84\\26&87\end{bmatrix}$, $\begin{bmatrix}110&59\\41&32\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 112.48.0.bv.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $8$ |
| 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 |
|---|---|---|---|---|---|
| 16.48.0-16.g.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.bv.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 112.48.0-112.f.2.2 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-112.f.2.13 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-16.g.1.1 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bv.1.1 | $112$ | $2$ | $2$ | $0$ | $?$ |
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.l.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.t.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.bo.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.bs.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dl.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dq.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.ec.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.ef.1.3 | $112$ | $2$ | $2$ | $1$ |
| 224.192.1-224.d.2.8 | $224$ | $2$ | $2$ | $1$ |
| 224.192.1-224.p.1.4 | $224$ | $2$ | $2$ | $1$ |
| 224.192.1-224.t.2.4 | $224$ | $2$ | $2$ | $1$ |
| 224.192.1-224.x.1.2 | $224$ | $2$ | $2$ | $1$ |
| 224.192.3-224.bc.1.16 | $224$ | $2$ | $2$ | $3$ |
| 224.192.3-224.bg.1.15 | $224$ | $2$ | $2$ | $3$ |
| 224.192.3-224.bw.1.15 | $224$ | $2$ | $2$ | $3$ |
| 224.192.3-224.ci.1.13 | $224$ | $2$ | $2$ | $3$ |