Invariants
Level: | $112$ | $\SL_2$-level: | $112$ | Newform level: | $1$ | ||
Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
Genus: | $13 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (all of which are rational) | Cusp widths | $2^{2}\cdot4\cdot14^{2}\cdot16\cdot28\cdot112$ | Cusp orbits | $1^{8}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $4 \le \gamma \le 13$ | ||||||
$\overline{\Q}$-gonality: | $4 \le \gamma \le 13$ | ||||||
Rational cusps: | $8$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 112D13 |
Level structure
$\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}2&29\\57&30\end{bmatrix}$, $\begin{bmatrix}3&0\\68&47\end{bmatrix}$, $\begin{bmatrix}19&34\\8&73\end{bmatrix}$, $\begin{bmatrix}74&5\\65&70\end{bmatrix}$, $\begin{bmatrix}82&11\\67&82\end{bmatrix}$, $\begin{bmatrix}82&15\\9&4\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 112.192.13.g.1 for the level structure with $-I$) |
Cyclic 112-isogeny field degree: | $2$ |
Cyclic 112-torsion field degree: | $48$ |
Full 112-torsion field degree: | $129024$ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
$X_0(7)$ | $7$ | $48$ | $24$ | $0$ | $0$ |
16.48.1-16.a.1.10 | $16$ | $8$ | $8$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
16.48.1-16.a.1.10 | $16$ | $8$ | $8$ | $1$ | $0$ |
56.192.5-56.bl.1.47 | $56$ | $2$ | $2$ | $5$ | $0$ |
112.192.5-56.bl.1.31 | $112$ | $2$ | $2$ | $5$ | $?$ |