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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}45&124\\68&57\end{bmatrix}$, $\begin{bmatrix}65&146\\40&111\end{bmatrix}$, $\begin{bmatrix}89&164\\44&33\end{bmatrix}$, $\begin{bmatrix}103&60\\12&79\end{bmatrix}$, $\begin{bmatrix}149&58\\136&117\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.48.0.bf.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $3072$ |
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 |
---|---|---|---|---|---|
24.48.0-8.d.2.9 | $24$ | $2$ | $2$ | $0$ | $0$ |
56.48.0-8.d.2.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
168.48.0-168.e.1.8 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.e.1.28 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.t.1.13 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.48.0-168.t.1.49 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.d.2.15 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.bf.2.10 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ei.2.14 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.eo.2.16 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.if.2.9 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ih.2.7 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ka.2.8 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.kc.2.13 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.mb.2.4 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.md.2.14 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.nw.2.16 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ny.2.12 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.pb.2.6 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.ph.2.2 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.py.2.10 | $168$ | $2$ | $2$ | $1$ |
168.192.1-168.qb.2.8 | $168$ | $2$ | $2$ | $1$ |
168.288.8-168.dk.1.23 | $168$ | $3$ | $3$ | $8$ |
168.384.7-168.dg.1.23 | $168$ | $4$ | $4$ | $7$ |