Invariants
Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $12^{12}$ | Cusp orbits | $4^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $4 \le \gamma \le 12$ | ||||||
$\overline{\Q}$-gonality: | $4 \le \gamma \le 7$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12B7 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}91&90\\18&85\end{bmatrix}$, $\begin{bmatrix}101&50\\32&133\end{bmatrix}$, $\begin{bmatrix}107&106\\112&157\end{bmatrix}$, $\begin{bmatrix}121&106\\30&119\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.144.7.kl.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $128$ |
Cyclic 168-torsion field degree: | $6144$ |
Full 168-torsion field degree: | $516096$ |
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 |
---|---|---|---|---|---|
24.144.3-24.ba.1.5 | $24$ | $2$ | $2$ | $3$ | $1$ |
84.144.4-84.z.1.8 | $84$ | $2$ | $2$ | $4$ | $?$ |
168.144.3-24.ba.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.144.3-168.dh.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.144.3-168.dh.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.144.3-168.di.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.144.3-168.di.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ |
168.144.4-168.i.1.6 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.i.1.31 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-84.z.1.14 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.ca.1.4 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.ca.1.27 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.dh.1.3 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.dh.1.26 | $168$ | $2$ | $2$ | $4$ | $?$ |