Invariants
| Level: | $224$ | $\SL_2$-level: | $32$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $2^{8}\cdot4^{12}\cdot32^{4}$ | Cusp orbits | $2^{4}\cdot4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 32N5 |
Level structure
| $\GL_2(\Z/224\Z)$-generators: | $\begin{bmatrix}63&32\\1&57\end{bmatrix}$, $\begin{bmatrix}189&72\\74&205\end{bmatrix}$, $\begin{bmatrix}195&8\\41&69\end{bmatrix}$, $\begin{bmatrix}199&24\\11&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 224.192.5.bd.2 for the level structure with $-I$) |
| Cyclic 224-isogeny field degree: | $32$ |
| Cyclic 224-torsion field degree: | $768$ |
| Full 224-torsion field degree: | $2064384$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=31$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 32.192.2-32.d.1.4 | $32$ | $2$ | $2$ | $2$ | $0$ |
| 112.192.1-112.bf.2.2 | $112$ | $2$ | $2$ | $1$ | $?$ |
| 224.192.1-112.bf.2.3 | $224$ | $2$ | $2$ | $1$ | $?$ |
| 224.192.2-224.c.1.6 | $224$ | $2$ | $2$ | $2$ | $?$ |
| 224.192.2-224.c.1.32 | $224$ | $2$ | $2$ | $2$ | $?$ |
| 224.192.2-32.d.1.15 | $224$ | $2$ | $2$ | $2$ | $?$ |