Invariants
Level: | $168$ | $\SL_2$-level: | $12$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12E0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}54&83\\85&128\end{bmatrix}$, $\begin{bmatrix}89&142\\50&129\end{bmatrix}$, $\begin{bmatrix}94&23\\45&110\end{bmatrix}$, $\begin{bmatrix}147&92\\20&159\end{bmatrix}$, $\begin{bmatrix}153&140\\46&131\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.24.0.fn.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $32$ |
Cyclic 168-torsion field degree: | $768$ |
Full 168-torsion field degree: | $3096576$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.24.0-6.a.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
168.24.0-6.a.1.3 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.96.1-168.dj.1.25 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.gj.1.3 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.kc.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.ke.1.4 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.bku.1.7 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.bkw.1.13 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.bld.1.5 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.blf.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.byi.1.5 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.byk.1.1 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.byr.1.5 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.byt.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.bzs.1.7 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.bzu.1.15 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.cab.1.3 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.cad.1.8 | $168$ | $2$ | $2$ | $1$ |
168.144.1-168.ca.1.4 | $168$ | $3$ | $3$ | $1$ |
168.384.11-168.sd.1.16 | $168$ | $8$ | $8$ | $11$ |