Invariants
Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $2^{6}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 24V3 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}41&39\\32&43\end{bmatrix}$, $\begin{bmatrix}43&117\\36&109\end{bmatrix}$, $\begin{bmatrix}53&51\\60&119\end{bmatrix}$, $\begin{bmatrix}105&86\\92&141\end{bmatrix}$, $\begin{bmatrix}115&105\\4&53\end{bmatrix}$, $\begin{bmatrix}147&151\\104&157\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.96.3.hh.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $16$ |
Cyclic 168-torsion field degree: | $768$ |
Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
$X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
56.48.0-56.u.1.4 | $56$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.96.1-12.h.1.23 | $24$ | $2$ | $2$ | $1$ | $0$ |
56.48.0-56.u.1.4 | $56$ | $4$ | $4$ | $0$ | $0$ |
168.96.1-12.h.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.1-168.zt.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.1-168.zt.1.29 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.1-168.zt.1.36 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.1-168.zt.1.61 | $168$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.