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: | $\begin{bmatrix}13&28\\88&159\end{bmatrix}$, $\begin{bmatrix}67&152\\151&31\end{bmatrix}$, $\begin{bmatrix}95&76\\160&97\end{bmatrix}$, $\begin{bmatrix}157&136\\159&97\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.24.0.bm.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $3072$ |
Full 168-torsion field degree: | $3096576$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 7 x^{2} + 16 y^{2} + 7 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.o.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ |
84.24.0-28.g.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-28.g.1.5 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-8.o.1.8 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-56.z.1.4 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-56.z.1.14 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.ka.1.30 | $168$ | $3$ | $3$ | $4$ |
168.192.3-168.mi.1.31 | $168$ | $4$ | $4$ | $3$ |
168.384.11-56.eo.1.1 | $168$ | $8$ | $8$ | $11$ |