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: | $12$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 24W3 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}49&122\\138&113\end{bmatrix}$, $\begin{bmatrix}55&140\\120&71\end{bmatrix}$, $\begin{bmatrix}74&105\\77&130\end{bmatrix}$, $\begin{bmatrix}122&81\\41&142\end{bmatrix}$, $\begin{bmatrix}157&150\\120&79\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.96.3.pw.2 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 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.96.0-12.c.3.3 | $12$ | $2$ | $2$ | $0$ | $0$ |
168.96.0-12.c.3.10 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.96.1-168.zs.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.1-168.zs.1.21 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.2-168.g.1.7 | $168$ | $2$ | $2$ | $2$ | $?$ |
168.96.2-168.g.1.17 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.384.5-168.mm.4.23 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.rn.1.4 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.sk.1.2 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.st.1.2 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.wa.1.2 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.wl.1.4 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.xo.1.4 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.xz.1.4 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.zj.3.4 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.zq.1.4 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.baq.4.15 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.bav.1.4 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.bcd.1.2 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.bck.1.2 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.bcu.1.1 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.bcz.1.4 | $168$ | $2$ | $2$ | $5$ |