Invariants
Level: | $168$ | $\SL_2$-level: | $8$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}47&164\\108&151\end{bmatrix}$, $\begin{bmatrix}49&120\\67&103\end{bmatrix}$, $\begin{bmatrix}51&160\\23&105\end{bmatrix}$, $\begin{bmatrix}111&16\\146&71\end{bmatrix}$, $\begin{bmatrix}111&68\\58&15\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.24.0.bf.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $3072$ |
Full 168-torsion field degree: | $3096576$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.24.0-4.d.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-4.d.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.96.1-168.ky.1.2 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.kz.1.4 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.la.1.2 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.lb.1.5 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.lc.1.1 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.ld.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.le.1.4 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.lf.1.3 | $168$ | $2$ | $2$ | $1$ |
168.144.4-168.fw.1.12 | $168$ | $3$ | $3$ | $4$ |
168.192.3-168.hk.1.4 | $168$ | $4$ | $4$ | $3$ |
168.384.11-168.ey.1.25 | $168$ | $8$ | $8$ | $11$ |