Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12A4 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}41&8\\78&1\end{bmatrix}$, $\begin{bmatrix}41&114\\66&83\end{bmatrix}$, $\begin{bmatrix}73&110\\80&83\end{bmatrix}$, $\begin{bmatrix}105&26\\80&95\end{bmatrix}$, $\begin{bmatrix}127&80\\110&59\end{bmatrix}$, $\begin{bmatrix}151&4\\108&107\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.f.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $128$ |
| Cyclic 168-torsion field degree: | $6144$ |
| Full 168-torsion field degree: | $1032192$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 6 x^{2} - z^{2} - w^{2} $ |
| $=$ | $ - x z w + 24 y^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 54 x^{4} z^{2} + x^{2} z^{4} + y^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{2}-zw+w^{2})^{3}(z^{2}+zw+w^{2})^{3}}{w^{4}z^{4}(z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 6y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Equation of the image curve:
| $0$ | $=$ | $ -54X^{4}Z^{2}+X^{2}Z^{4}+Y^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 84.72.2-12.a.1.8 | $84$ | $2$ | $2$ | $2$ | $?$ |
| 168.48.0-24.c.1.4 | $168$ | $3$ | $3$ | $0$ | $?$ |
| 168.72.2-12.a.1.8 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-24.d.1.2 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-24.d.1.11 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-24.d.1.14 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.