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: | $\begin{bmatrix}21&64\\106&99\end{bmatrix}$, $\begin{bmatrix}27&20\\110&11\end{bmatrix}$, $\begin{bmatrix}57&80\\118&69\end{bmatrix}$, $\begin{bmatrix}101&64\\32&129\end{bmatrix}$, $\begin{bmatrix}163&124\\84&85\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.48.0.cr.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.m.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
56.48.0-56.i.2.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
168.48.0-56.i.2.3 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-24.m.1.19 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.t.2.4 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.t.2.29 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.x.2.4 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.bc.2.3 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.cj.2.4 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.cm.2.3 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.fi.2.12 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.fj.2.7 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.fm.2.8 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.fn.2.7 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ng.1.4 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.nh.1.2 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ns.1.4 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.nt.1.2 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.nw.1.12 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.nx.1.6 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.oi.1.8 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.oj.1.6 | $168$ | $2$ | $2$ | $1$ |
168.288.8-168.ol.1.49 | $168$ | $3$ | $3$ | $8$ |
168.384.7-168.ja.1.49 | $168$ | $4$ | $4$ | $7$ |