Invariants
| Level: | $168$ | $\SL_2$-level: | $42$ | Newform level: | $1$ | ||
| Index: | $336$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $12 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $14^{3}\cdot42^{3}$ | Cusp orbits | $3^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 22$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 12$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 42C12 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}4&51\\105&31\end{bmatrix}$, $\begin{bmatrix}46&155\\97&45\end{bmatrix}$, $\begin{bmatrix}71&127\\125&60\end{bmatrix}$, $\begin{bmatrix}111&133\\7&48\end{bmatrix}$, $\begin{bmatrix}151&5\\10&129\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.168.12.c.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $96$ |
| Cyclic 168-torsion field degree: | $4608$ |
| Full 168-torsion field degree: | $442368$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $16$ | $8$ | $0$ | $0$ |
| 24.16.0-24.c.1.6 | $24$ | $21$ | $21$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 84.168.5-21.a.1.13 | $84$ | $2$ | $2$ | $5$ | $?$ |
| 168.168.5-21.a.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ |
| 24.16.0-24.c.1.6 | $24$ | $21$ | $21$ | $0$ | $0$ |