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 | $1^{2}\cdot2\cdot4\cdot8^{2}$ | 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: | 8I0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}38&81\\57&110\end{bmatrix}$, $\begin{bmatrix}63&50\\58&23\end{bmatrix}$, $\begin{bmatrix}80&3\\7&92\end{bmatrix}$, $\begin{bmatrix}93&78\\84&87\end{bmatrix}$, $\begin{bmatrix}97&82\\94&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.bu.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $384$ |
| 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 74 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{7^2\cdot13^8}\cdot\frac{(7x+y)^{24}(62121073x^{8}-51323776x^{7}y-226174200x^{6}y^{2}-160864256x^{5}y^{3}+61244120x^{4}y^{4}+183318016x^{3}y^{5}+109082400x^{2}y^{6}+22378496xy^{7}+1811728y^{8})^{3}}{(x+2y)^{4}(7x+y)^{26}(7x^{2}-2y^{2})^{8}(42x^{2}-14xy-27y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 112.24.0-8.n.1.6 | $112$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 112.96.0-56.z.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bc.2.7 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bd.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.be.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bg.2.8 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bj.1.8 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bl.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bm.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.384.11-56.fc.1.25 | $112$ | $8$ | $8$ | $11$ |
| 112.96.0-112.bc.2.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bi.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bk.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bq.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bs.2.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bu.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bw.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.by.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.bg.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bi.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bk.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bm.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bo.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bu.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bw.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.cc.2.2 | $112$ | $2$ | $2$ | $1$ |