Invariants
| Level: | $168$ | $\SL_2$-level: | $28$ | Newform level: | $1$ | ||
| Index: | $336$ | $\PSL_2$-index: | $336$ | ||||
| Genus: | $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $14^{8}\cdot28^{8}$ | Cusp orbits | $2^{2}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 40$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 21$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28E21 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}25&84\\102&17\end{bmatrix}$, $\begin{bmatrix}28&39\\101&158\end{bmatrix}$, $\begin{bmatrix}60&55\\113&164\end{bmatrix}$, $\begin{bmatrix}65&68\\84&103\end{bmatrix}$, $\begin{bmatrix}112&125\\69&140\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $442368$ |
Rational points
This modular curve has no $\Q_p$ points for $p=17,19,43$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.168.11.a.1 | $56$ | $2$ | $2$ | $11$ | $1$ |
| 84.168.9.bg.1 | $84$ | $2$ | $2$ | $9$ | $?$ |
| 168.168.7.b.1 | $168$ | $2$ | $2$ | $7$ | $?$ |