Invariants
Level: | $112$ | $\SL_2$-level: | $112$ | Newform level: | $1$ | ||
Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
Genus: | $11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $1^{4}\cdot4\cdot7^{4}\cdot16\cdot28\cdot112$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $5 \le \gamma \le 11$ | ||||||
$\overline{\Q}$-gonality: | $5 \le \gamma \le 11$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 112H11 |
Level structure
$\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}7&22\\92&105\end{bmatrix}$, $\begin{bmatrix}43&2\\94&63\end{bmatrix}$, $\begin{bmatrix}52&7\\43&16\end{bmatrix}$, $\begin{bmatrix}77&82\\10&93\end{bmatrix}$, $\begin{bmatrix}96&109\\93&56\end{bmatrix}$, $\begin{bmatrix}104&91\\25&2\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 112.192.11.z.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 4 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 |
---|---|---|---|---|---|
$X_0(7)$ | $7$ | $48$ | $24$ | $0$ | $0$ |
16.48.0-16.h.1.14 | $16$ | $8$ | $8$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
16.48.0-16.h.1.14 | $16$ | $8$ | $8$ | $0$ | $0$ |
56.192.5-56.bl.1.2 | $56$ | $2$ | $2$ | $5$ | $0$ |
112.192.5-56.bl.1.31 | $112$ | $2$ | $2$ | $5$ | $?$ |