Invariants
Level: | $168$ | $\SL_2$-level: | $24$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $6$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12E0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}38&133\\105&142\end{bmatrix}$, $\begin{bmatrix}49&120\\46&155\end{bmatrix}$, $\begin{bmatrix}70&135\\25&68\end{bmatrix}$, $\begin{bmatrix}112&131\\109&90\end{bmatrix}$, $\begin{bmatrix}115&42\\0&157\end{bmatrix}$, $\begin{bmatrix}124&89\\85&12\end{bmatrix}$, $\begin{bmatrix}141&110\\64&167\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $16$ |
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, including 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
168.12.0-4.c.1.5 | $168$ | $4$ | $4$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.