Invariants
Level: | $168$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-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: | $\begin{bmatrix}15&44\\62&155\end{bmatrix}$, $\begin{bmatrix}23&0\\115&41\end{bmatrix}$, $\begin{bmatrix}25&136\\147&163\end{bmatrix}$, $\begin{bmatrix}47&132\\144&95\end{bmatrix}$, $\begin{bmatrix}137&20\\55&35\end{bmatrix}$, $\begin{bmatrix}157&96\\63&131\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 168.48.0-168.bf.1.1, 168.48.0-168.bf.1.2, 168.48.0-168.bf.1.3, 168.48.0-168.bf.1.4, 168.48.0-168.bf.1.5, 168.48.0-168.bf.1.6, 168.48.0-168.bf.1.7, 168.48.0-168.bf.1.8, 168.48.0-168.bf.1.9, 168.48.0-168.bf.1.10, 168.48.0-168.bf.1.11, 168.48.0-168.bf.1.12, 168.48.0-168.bf.1.13, 168.48.0-168.bf.1.14, 168.48.0-168.bf.1.15, 168.48.0-168.bf.1.16 |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $3072$ |
Full 168-torsion field degree: | $6193152$ |
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 |
---|---|---|---|---|---|
4.12.0.d.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
168.12.0.y.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.12.0.ek.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.48.1.ky.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.kz.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.la.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.lb.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.lc.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.ld.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.le.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.lf.1 | $168$ | $2$ | $2$ | $1$ |
168.72.4.fw.1 | $168$ | $3$ | $3$ | $4$ |
168.96.3.hk.1 | $168$ | $4$ | $4$ | $3$ |
168.192.11.ey.1 | $168$ | $8$ | $8$ | $11$ |