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: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G0 |
Level structure
$\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}19&36\\21&109\end{bmatrix}$, $\begin{bmatrix}39&64\\103&111\end{bmatrix}$, $\begin{bmatrix}41&52\\57&15\end{bmatrix}$, $\begin{bmatrix}107&88\\31&95\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.24.0.bm.1 for the level structure with $-I$) |
Cyclic 112-isogeny field degree: | $32$ |
Cyclic 112-torsion field degree: | $1536$ |
Full 112-torsion field degree: | $1032192$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 7 x^{2} + 16 y^{2} + 7 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
16.24.0-8.o.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ |
112.24.0-8.o.1.3 | $112$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
112.96.1-112.y.1.4 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.y.1.8 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.ba.1.4 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.ba.1.8 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.cm.1.4 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.cm.1.8 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.co.1.4 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.co.1.8 | $112$ | $2$ | $2$ | $1$ |
112.384.11-56.eo.1.8 | $112$ | $8$ | $8$ | $11$ |