Invariants
Level: | $168$ | $\SL_2$-level: | $8$ | ||||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
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: | 8O0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}9&52\\130&63\end{bmatrix}$, $\begin{bmatrix}23&120\\158&161\end{bmatrix}$, $\begin{bmatrix}125&124\\62&167\end{bmatrix}$, $\begin{bmatrix}143&36\\78&77\end{bmatrix}$, $\begin{bmatrix}165&64\\76&155\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.48.0.cf.2 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $1536$ |
Full 168-torsion field degree: | $1548288$ |
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 |
---|---|---|---|---|---|
24.48.0-24.i.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
56.48.0-56.l.1.8 | $56$ | $2$ | $2$ | $0$ | $0$ |
168.48.0-24.i.2.16 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-56.l.1.9 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.t.2.4 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.t.2.16 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.192.1-168.y.1.4 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.bg.1.2 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.cq.1.6 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.cx.1.3 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.dm.1.2 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.dn.1.1 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.dq.1.8 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.dr.1.7 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ig.1.6 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ih.1.4 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.iq.1.6 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ir.1.4 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.jm.1.8 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.jn.1.8 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.jw.1.3 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.jx.1.3 | $168$ | $2$ | $2$ | $1$ |
168.288.8-168.nc.2.50 | $168$ | $3$ | $3$ | $8$ |
168.384.7-168.hk.2.7 | $168$ | $4$ | $4$ | $7$ |