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: | $\begin{bmatrix}11&32\\155&93\end{bmatrix}$, $\begin{bmatrix}79&92\\47&9\end{bmatrix}$, $\begin{bmatrix}97&156\\121&163\end{bmatrix}$, $\begin{bmatrix}107&8\\64&107\end{bmatrix}$, $\begin{bmatrix}141&128\\50&1\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.24.0.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: | $3096576$ |
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 |
---|---|---|---|---|---|
12.24.0-4.d.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-4.d.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ |
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.me.1.5 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.mf.1.8 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.mg.1.3 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.mh.1.4 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.mi.1.2 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.mj.1.8 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.mk.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.ml.1.2 | $168$ | $2$ | $2$ | $1$ |
168.144.4-168.hc.1.16 | $168$ | $3$ | $3$ | $4$ |
168.192.3-168.iq.1.4 | $168$ | $4$ | $4$ | $3$ |
168.384.11-168.ge.1.29 | $168$ | $8$ | $8$ | $11$ |