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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | 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: | 16H0 |
Level structure
$\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}4&21\\81&48\end{bmatrix}$, $\begin{bmatrix}18&25\\57&2\end{bmatrix}$, $\begin{bmatrix}55&94\\44&49\end{bmatrix}$, $\begin{bmatrix}101&84\\58&71\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 112.48.0.bk.2 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 real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
16.48.0-16.e.2.1 | $16$ | $2$ | $2$ | $0$ | $0$ |
56.48.0-56.bu.2.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
112.48.0-16.e.2.13 | $112$ | $2$ | $2$ | $0$ | $?$ |
112.48.0-112.h.1.6 | $112$ | $2$ | $2$ | $0$ | $?$ |
112.48.0-112.h.1.9 | $112$ | $2$ | $2$ | $0$ | $?$ |
112.48.0-56.bu.2.5 | $112$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
112.192.1-112.m.1.1 | $112$ | $2$ | $2$ | $1$ |
112.192.1-112.y.2.1 | $112$ | $2$ | $2$ | $1$ |
112.192.1-112.bf.1.2 | $112$ | $2$ | $2$ | $1$ |
112.192.1-112.bz.1.1 | $112$ | $2$ | $2$ | $1$ |
112.192.1-112.ce.1.1 | $112$ | $2$ | $2$ | $1$ |
112.192.1-112.cr.2.1 | $112$ | $2$ | $2$ | $1$ |
112.192.1-112.cv.1.1 | $112$ | $2$ | $2$ | $1$ |
112.192.1-112.dg.1.1 | $112$ | $2$ | $2$ | $1$ |