Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}47&88\\34&19\end{bmatrix}$, $\begin{bmatrix}65&8\\28&55\end{bmatrix}$, $\begin{bmatrix}97&16\\91&57\end{bmatrix}$, $\begin{bmatrix}97&40\\8&59\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.bl.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $384$ |
| Full 112-torsion field degree: | $516096$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^2\cdot7}\cdot\frac{(3x+2y)^{48}(2777073111873x^{16}+8637353266464x^{15}y+8204207593824x^{14}y^{2}-32855808384x^{13}y^{3}-5365496360448x^{12}y^{4}-4256335060992x^{11}y^{5}-1188946418688x^{10}y^{6}-112789573632x^{9}y^{7}+624669345792x^{8}y^{8}+988886237184x^{7}y^{9}+670365351936x^{6}y^{10}+255281725440x^{5}y^{11}+61535158272x^{4}y^{12}+10305404928x^{3}y^{13}+1028653056x^{2}y^{14}+12582912xy^{15}+1048576y^{16})^{3}}{x^{2}(3x+2y)^{48}(3x+4y)^{2}(9x^{2}+3xy+2y^{2})^{4}(27x^{2}-12xy-8y^{2})^{2}(81x^{4}-1080x^{3}y-648x^{2}y^{2}+96xy^{3}+32y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.bb.2.7 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 112.48.0-8.bb.2.8 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bj.1.4 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bj.1.6 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.1.1 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.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 |
|---|---|---|---|---|
| 112.192.1-112.cv.2.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.cx.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dd.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.df.2.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.eb.1.9 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.ed.2.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.ej.2.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.el.1.5 | $112$ | $2$ | $2$ | $1$ |