Invariants
Level: | $168$ | $\SL_2$-level: | $8$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $192$ | ||||
Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
Cusps: | $24$ (none of which are rational) | Cusp widths | $8^{24}$ | Cusp orbits | $2^{2}\cdot4^{3}\cdot8$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8A5 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}1&108\\52&49\end{bmatrix}$, $\begin{bmatrix}71&88\\92&167\end{bmatrix}$, $\begin{bmatrix}111&26\\20&133\end{bmatrix}$, $\begin{bmatrix}149&48\\40&25\end{bmatrix}$, $\begin{bmatrix}157&144\\76&121\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 168.384.5-168.ew.2.1, 168.384.5-168.ew.2.2, 168.384.5-168.ew.2.3, 168.384.5-168.ew.2.4, 168.384.5-168.ew.2.5, 168.384.5-168.ew.2.6, 168.384.5-168.ew.2.7, 168.384.5-168.ew.2.8, 168.384.5-168.ew.2.9, 168.384.5-168.ew.2.10, 168.384.5-168.ew.2.11, 168.384.5-168.ew.2.12, 168.384.5-168.ew.2.13, 168.384.5-168.ew.2.14, 168.384.5-168.ew.2.15, 168.384.5-168.ew.2.16 |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $3072$ |
Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has no $\Q_p$ points for $p=29$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.96.3.u.2 | $24$ | $2$ | $2$ | $3$ | $0$ |
56.96.1.n.2 | $56$ | $2$ | $2$ | $1$ | $1$ |
168.96.1.bw.2 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.1.ca.2 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.3.be.1 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.96.3.bo.3 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.96.3.ca.1 | $168$ | $2$ | $2$ | $3$ | $?$ |