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: | $2$ | ||||||
$\overline{\Q}$-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: | $\begin{bmatrix}25&158\\52&117\end{bmatrix}$, $\begin{bmatrix}43&122\\118&129\end{bmatrix}$, $\begin{bmatrix}139&110\\132&85\end{bmatrix}$, $\begin{bmatrix}153&130\\10&67\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.24.0.g.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
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 2 x^{2} - 3 x y + 2 y^{2} + 56 z^{2} $ |
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 |
---|---|---|---|---|---|
24.24.0-8.b.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
84.24.0-28.b.1.1 | $84$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-56.a.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-56.a.1.7 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-8.b.1.3 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-28.b.1.4 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.g.1.27 | $168$ | $3$ | $3$ | $4$ |
168.192.3-168.cw.1.19 | $168$ | $4$ | $4$ | $3$ |
168.384.11-56.j.1.1 | $168$ | $8$ | $8$ | $11$ |