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 | $2^{5}$ | ||
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: | $\begin{bmatrix}73&112\\10&123\end{bmatrix}$, $\begin{bmatrix}85&56\\132&61\end{bmatrix}$, $\begin{bmatrix}89&24\\6&113\end{bmatrix}$, $\begin{bmatrix}121&56\\0&13\end{bmatrix}$, $\begin{bmatrix}121&80\\78&139\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.48.0.cw.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $32$ |
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 |
---|---|---|---|---|---|
8.48.0-8.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
168.48.0-8.i.1.3 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.t.1.3 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.t.1.29 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.ec.1.13 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.ec.1.20 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.v.1.2 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.fo.1.7 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.kx.1.2 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.lb.1.4 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ot.2.3 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ox.2.13 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.pt.2.3 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.qb.2.7 | $168$ | $2$ | $2$ | $1$ |
168.288.8-168.pb.2.41 | $168$ | $3$ | $3$ | $8$ |
168.384.7-168.jo.2.1 | $168$ | $4$ | $4$ | $7$ |