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: | $\begin{bmatrix}9&54\\14&57\end{bmatrix}$, $\begin{bmatrix}40&27\\67&48\end{bmatrix}$, $\begin{bmatrix}51&12\\86&105\end{bmatrix}$, $\begin{bmatrix}61&54\\56&107\end{bmatrix}$, $\begin{bmatrix}104&79\\15&88\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.bv.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 63 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3\cdot7^2}\cdot\frac{(x-3y)^{24}(479x^{8}+7872x^{7}y+27216x^{6}y^{2}+18144x^{5}y^{3}-108864x^{3}y^{5}-163296x^{2}y^{6}-139968xy^{7}-104976y^{8})^{3}}{x^{4}(x-3y)^{24}(x+6y)^{2}(2x^{2}+3xy+9y^{2})^{8}(5x^{2}+4xy+12y^{2})}$ |
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.1.5 | $112$ | $2$ | $2$ | $0$ | $?$ |
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.bb.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bc.2.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bd.2.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bf.1.7 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bh.2.8 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bi.2.8 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bk.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bn.2.6 | $112$ | $2$ | $2$ | $0$ |
| 112.384.11-56.fd.2.25 | $112$ | $8$ | $8$ | $11$ |
| 112.96.0-112.bd.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bj.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bl.2.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.br.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bt.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bv.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bx.2.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bz.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.bh.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bj.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bl.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bn.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bp.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bv.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bx.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.cd.1.5 | $112$ | $2$ | $2$ | $1$ |