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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}9&32\\23&73\end{bmatrix}$, $\begin{bmatrix}23&104\\88&7\end{bmatrix}$, $\begin{bmatrix}27&8\\73&23\end{bmatrix}$, $\begin{bmatrix}89&24\\69&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.bj.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $16$ |
| Cyclic 112-torsion field degree: | $192$ |
| 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 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{7}\cdot\frac{(x+4y)^{48}(73201x^{16}+4535328x^{15}y+126530208x^{14}y^{2}+2083890816x^{13}y^{3}+22195475904x^{12}y^{4}+155738903040x^{11}y^{5}+684636341760x^{10}y^{6}+1579720009728x^{9}y^{7}+2423911917056x^{8}y^{8}+44232160272384x^{7}y^{9}+536754891939840x^{6}y^{10}+3418780399534080x^{5}y^{11}+13642582437249024x^{4}y^{12}+35864527815180288x^{3}y^{13}+60973680398303232x^{2}y^{14}+61194856482471936xy^{15}+27655484080193536y^{16})^{3}}{(x+4y)^{48}(x^{2}-28y^{2})^{2}(x^{2}+12xy+28y^{2})^{2}(x^{2}+28xy+140y^{2})^{8}(3x^{2}+28xy+84y^{2})^{4}(5x^{2}+28xy+28y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.bb.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 112.48.0-8.bb.1.4 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bh.1.4 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bh.1.5 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.1.1 | $112$ | $2$ | $2$ | $0$ | $?$ |
| 112.48.0-56.bu.1.13 | $112$ | $2$ | $2$ | $0$ | $?$ |
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.cj.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.cl.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.cr.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.ct.1.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dp.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dr.2.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dx.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.dz.1.1 | $112$ | $2$ | $2$ | $1$ |