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: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-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: | $\begin{bmatrix}47&128\\150&29\end{bmatrix}$, $\begin{bmatrix}93&128\\44&161\end{bmatrix}$, $\begin{bmatrix}129&40\\143&77\end{bmatrix}$, $\begin{bmatrix}161&164\\134&141\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.24.0.bx.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $1536$ |
Full 168-torsion field degree: | $3096576$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
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
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.24.0-24.s.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
168.24.0-24.s.1.5 | $168$ | $2$ | $2$ | $0$ | $?$ |
56.24.0-56.bb.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ |
168.24.0-56.bb.1.4 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-168.z.1.13 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-168.z.1.30 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.144.4-168.hz.1.2 | $168$ | $3$ | $3$ | $4$ |
168.192.3-168.jv.1.2 | $168$ | $4$ | $4$ | $3$ |
168.384.11-168.hv.1.11 | $168$ | $8$ | $8$ | $11$ |