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 $4$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}29&0\\62&37\end{bmatrix}$, $\begin{bmatrix}31&16\\66&87\end{bmatrix}$, $\begin{bmatrix}47&0\\48&19\end{bmatrix}$, $\begin{bmatrix}69&88\\74&63\end{bmatrix}$, $\begin{bmatrix}93&8\\60&7\end{bmatrix}$, $\begin{bmatrix}97&24\\40&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.i.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, including 122 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}+240x^{6}y^{2}+2144x^{4}y^{4}+3840x^{2}y^{6}+256y^{8})^{3}}{y^{2}x^{26}(x-2y)^{8}(x+2y)^{8}(x^{2}+4y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 112.24.0-8.n.1.3 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.24.0-8.n.1.7 | $112$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 112.96.0-16.d.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-16.d.1.12 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-16.d.2.10 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-16.d.2.12 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.d.1.15 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.d.1.24 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.d.2.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.d.2.16 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-8.j.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-8.j.2.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-8.k.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-8.k.2.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-8.l.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-8.l.2.6 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.z.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.z.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.ba.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.ba.2.6 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bb.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bb.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-16.a.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.a.1.9 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.a.2.9 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.a.1.15 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.a.2.4 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.a.2.16 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.b.1.4 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.b.1.12 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.b.2.11 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.b.1.15 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.b.2.4 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.b.2.16 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-8.h.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-8.p.1.4 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-56.bu.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-56.bv.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.96.2-16.d.1.7 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2-16.d.2.6 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2-112.d.1.19 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2-112.d.2.14 | $112$ | $2$ | $2$ | $2$ |
| 112.384.11-56.bn.1.12 | $112$ | $8$ | $8$ | $11$ |