Invariants
Level: | $168$ | $\SL_2$-level: | $4$ | ||||
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 | $4^{6}$ | 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: | 4G0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}25&72\\70&145\end{bmatrix}$, $\begin{bmatrix}59&116\\112&15\end{bmatrix}$, $\begin{bmatrix}97&84\\20&5\end{bmatrix}$, $\begin{bmatrix}109&166\\144&155\end{bmatrix}$, $\begin{bmatrix}165&2\\80&59\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.24.0.a.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $128$ |
Cyclic 168-torsion field degree: | $6144$ |
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 |
---|---|---|---|---|---|
24.24.0-4.a.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
28.24.0-4.a.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
168.24.0-168.a.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-168.a.1.7 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-168.a.1.15 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.96.1-168.f.1.7 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.h.1.7 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.cm.1.7 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.cp.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.em.1.8 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.ep.1.4 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.fh.1.8 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.fj.1.4 | $168$ | $2$ | $2$ | $1$ |
168.144.4-168.h.1.21 | $168$ | $3$ | $3$ | $4$ |
168.192.3-168.cx.1.13 | $168$ | $4$ | $4$ | $3$ |
168.384.11-168.k.1.15 | $168$ | $8$ | $8$ | $11$ |