Invariants
Level: | $168$ | $\SL_2$-level: | $28$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot14^{2}\cdot28^{2}$ | Cusp orbits | $2^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 28E5 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}15&28\\88&111\end{bmatrix}$, $\begin{bmatrix}23&154\\54&167\end{bmatrix}$, $\begin{bmatrix}33&14\\2&129\end{bmatrix}$, $\begin{bmatrix}55&154\\133&167\end{bmatrix}$, $\begin{bmatrix}145&56\\63&79\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.96.5.f.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
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
56.96.2-28.a.1.1 | $56$ | $2$ | $2$ | $2$ | $0$ |
168.24.0-84.c.1.3 | $168$ | $8$ | $8$ | $0$ | $?$ |
168.96.2-28.a.1.12 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.384.11-84.bb.1.6 | $168$ | $2$ | $2$ | $11$ |
168.384.11-84.bb.1.7 | $168$ | $2$ | $2$ | $11$ |
168.384.11-84.bb.2.5 | $168$ | $2$ | $2$ | $11$ |
168.384.11-84.bb.2.8 | $168$ | $2$ | $2$ | $11$ |
168.384.11-84.bc.1.3 | $168$ | $2$ | $2$ | $11$ |
168.384.11-84.bc.1.5 | $168$ | $2$ | $2$ | $11$ |
168.384.11-84.bc.2.1 | $168$ | $2$ | $2$ | $11$ |
168.384.11-84.bc.2.7 | $168$ | $2$ | $2$ | $11$ |
168.384.11-168.es.1.8 | $168$ | $2$ | $2$ | $11$ |
168.384.11-168.es.1.9 | $168$ | $2$ | $2$ | $11$ |
168.384.11-168.es.2.1 | $168$ | $2$ | $2$ | $11$ |
168.384.11-168.es.2.16 | $168$ | $2$ | $2$ | $11$ |
168.384.11-168.et.1.6 | $168$ | $2$ | $2$ | $11$ |
168.384.11-168.et.1.11 | $168$ | $2$ | $2$ | $11$ |
168.384.11-168.et.2.3 | $168$ | $2$ | $2$ | $11$ |
168.384.11-168.et.2.14 | $168$ | $2$ | $2$ | $11$ |