Invariants
Level: | $168$ | $\SL_2$-level: | $12$ | 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 | $4^{12}\cdot12^{12}$ | Cusp orbits | $2^{2}\cdot4^{5}$ | ||
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: | 12E5 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}51&16\\58&63\end{bmatrix}$, $\begin{bmatrix}87&164\\76&5\end{bmatrix}$, $\begin{bmatrix}135&124\\40&27\end{bmatrix}$, $\begin{bmatrix}147&146\\16&167\end{bmatrix}$, $\begin{bmatrix}159&62\\62&75\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 168.384.5-168.kv.2.1, 168.384.5-168.kv.2.2, 168.384.5-168.kv.2.3, 168.384.5-168.kv.2.4, 168.384.5-168.kv.2.5, 168.384.5-168.kv.2.6, 168.384.5-168.kv.2.7, 168.384.5-168.kv.2.8, 168.384.5-168.kv.2.9, 168.384.5-168.kv.2.10, 168.384.5-168.kv.2.11, 168.384.5-168.kv.2.12, 168.384.5-168.kv.2.13, 168.384.5-168.kv.2.14, 168.384.5-168.kv.2.15, 168.384.5-168.kv.2.16 |
Cyclic 168-isogeny field degree: | $32$ |
Cyclic 168-torsion field degree: | $768$ |
Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.96.1.cp.2 | $24$ | $2$ | $2$ | $1$ | $1$ |
84.96.3.p.2 | $84$ | $2$ | $2$ | $3$ | $?$ |
168.96.1.lq.2 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.1.ls.2 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.3.do.1 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.96.3.el.1 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.96.3.eu.1 | $168$ | $2$ | $2$ | $3$ | $?$ |