Invariants
Level: | $168$ | $\SL_2$-level: | $84$ | Newform level: | $1$ | ||
Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
Genus: | $13 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (all of which are rational) | Cusp widths | $2\cdot4\cdot6\cdot12\cdot14\cdot28\cdot42\cdot84$ | Cusp orbits | $1^{8}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $4 \le \gamma \le 13$ | ||||||
$\overline{\Q}$-gonality: | $4 \le \gamma \le 13$ | ||||||
Rational cusps: | $8$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 84G13 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}2&67\\39&16\end{bmatrix}$, $\begin{bmatrix}27&4\\38&49\end{bmatrix}$, $\begin{bmatrix}91&156\\12&109\end{bmatrix}$, $\begin{bmatrix}113&12\\104&7\end{bmatrix}$, $\begin{bmatrix}160&59\\141&92\end{bmatrix}$, $\begin{bmatrix}165&44\\98&153\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.192.13.pe.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $4$ |
Cyclic 168-torsion field degree: | $192$ |
Full 168-torsion field degree: | $387072$ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal 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 |
---|---|---|---|---|---|
$X_0(7)$ | $7$ | $48$ | $24$ | $0$ | $0$ |
24.48.1-24.es.1.4 | $24$ | $8$ | $8$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.48.1-24.es.1.4 | $24$ | $8$ | $8$ | $1$ | $0$ |
84.192.5-42.a.1.47 | $84$ | $2$ | $2$ | $5$ | $?$ |
168.192.5-42.a.1.54 | $168$ | $2$ | $2$ | $5$ | $?$ |