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: | $\begin{bmatrix}42&73\\111&92\end{bmatrix}$, $\begin{bmatrix}48&99\\43&8\end{bmatrix}$, $\begin{bmatrix}57&64\\20&53\end{bmatrix}$, $\begin{bmatrix}86&67\\51&14\end{bmatrix}$, $\begin{bmatrix}101&90\\58&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.bv.2 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{3^8\cdot5^8\cdot7^2}\cdot\frac{(7x+y)^{24}(112945441x^{8}+68534144x^{7}y-69168008x^{6}y^{2}+298898432x^{5}y^{3}-154983080x^{4}y^{4}+94857728x^{3}y^{5}-15780128x^{2}y^{6}-16758784xy^{7}-645104y^{8})^{3}}{(x-2y)^{4}(7x+y)^{26}(7x^{2}+2y^{2})^{8}(56x^{2}-14xy+29y^{2})}$ |
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.1.4 | $112$ | $2$ | $2$ | $0$ | $?$ |
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.2.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bc.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bd.1.7 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bf.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bh.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bi.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bk.2.8 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bn.1.8 | $112$ | $2$ | $2$ | $0$ |
| 112.384.11-56.fd.1.27 | $112$ | $8$ | $8$ | $11$ |
| 112.96.0-112.bd.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bj.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bl.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.br.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bt.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bv.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bx.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bz.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.bh.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bj.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bl.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bn.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bp.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bv.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bx.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.cd.2.1 | $112$ | $2$ | $2$ | $1$ |