Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{4}$ | ||
| 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: | 8C0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}20&39\\111&92\end{bmatrix}$, $\begin{bmatrix}51&80\\56&11\end{bmatrix}$, $\begin{bmatrix}54&81\\23&40\end{bmatrix}$, $\begin{bmatrix}70&57\\103&104\end{bmatrix}$, $\begin{bmatrix}89&78\\46&25\end{bmatrix}$, $\begin{bmatrix}103&36\\98&73\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $768$ |
| Full 112-torsion field degree: | $2064384$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 112.48.0-16.e.1.7 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.e.1.12 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.e.2.7 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.e.2.14 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.e.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.e.1.24 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.e.2.3 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.e.2.22 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.f.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.f.1.12 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.f.2.7 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.f.2.14 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.f.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.f.1.22 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.f.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.f.2.28 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.g.1.7 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.g.1.14 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.g.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.g.1.19 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.h.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.h.1.16 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.h.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.h.1.23 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.i.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.k.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.q.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.r.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.ba.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.ba.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.ba.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.ba.2.4 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.bb.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.bb.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.bb.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-8.bb.2.7 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bf.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bh.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bj.1.8 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bl.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bu.1.9 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bu.1.16 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bu.2.6 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bu.2.15 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bv.1.10 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bv.1.15 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bv.2.3 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-56.bv.2.16 | $112$ | $2$ | $2$ | $0$ |
| 112.48.1-16.a.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-16.a.1.16 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-112.a.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-112.a.1.23 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-16.b.1.7 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-16.b.1.14 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-112.b.1.6 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-112.b.1.19 | $112$ | $2$ | $2$ | $1$ |
| 112.192.5-56.bl.1.18 | $112$ | $8$ | $8$ | $5$ |
| 112.504.16-56.cj.1.14 | $112$ | $21$ | $21$ | $16$ |