Invariants
Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $384$ | $\PSL_2$-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: | $\begin{bmatrix}3&140\\88&113\end{bmatrix}$, $\begin{bmatrix}33&104\\140&3\end{bmatrix}$, $\begin{bmatrix}55&36\\162&121\end{bmatrix}$, $\begin{bmatrix}101&162\\2&109\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.192.5.ju.3 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $32$ |
Cyclic 168-torsion field degree: | $768$ |
Full 168-torsion field degree: | $387072$ |
Rational points
This modular curve has no $\Q_p$ points for $p=23$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.192.1-24.cl.1.9 | $24$ | $2$ | $2$ | $1$ | $0$ |
84.192.3-84.r.2.7 | $84$ | $2$ | $2$ | $3$ | $?$ |
168.192.1-24.cl.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.1-168.ls.3.8 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.1-168.ls.3.24 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.1-168.lw.2.2 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.1-168.lw.2.19 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.3-84.r.2.6 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.dd.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.dd.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.ef.1.4 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.ef.1.5 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.eu.1.9 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.eu.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ |