Invariants
Level: | $168$ | $\SL_2$-level: | $24$ | 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 | $2^{8}\cdot6^{8}\cdot8^{4}\cdot24^{4}$ | 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: | 24Z5 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}7&48\\53&133\end{bmatrix}$, $\begin{bmatrix}25&72\\2&67\end{bmatrix}$, $\begin{bmatrix}61&0\\33&23\end{bmatrix}$, $\begin{bmatrix}163&108\\153&43\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.192.5.bit.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $16$ |
Cyclic 168-torsion field degree: | $768$ |
Full 168-torsion field degree: | $387072$ |
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 |
---|---|---|---|---|---|
24.192.3-24.gm.3.8 | $24$ | $2$ | $2$ | $3$ | $0$ |
84.192.1-84.o.1.2 | $84$ | $2$ | $2$ | $1$ | $?$ |
168.192.1-84.o.1.17 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.1-168.sb.4.5 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.1-168.sb.4.18 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.1-168.sn.3.9 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.1-168.sn.3.18 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.192.3-24.gm.3.15 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.my.1.5 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.my.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.pe.1.4 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.pe.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.qy.3.9 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.192.3-168.qy.3.18 | $168$ | $2$ | $2$ | $3$ | $?$ |