Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $6^{4}\cdot24^{2}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D4 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}7&146\\158&57\end{bmatrix}$, $\begin{bmatrix}25&96\\78&7\end{bmatrix}$, $\begin{bmatrix}37&26\\134&47\end{bmatrix}$, $\begin{bmatrix}97&14\\164&165\end{bmatrix}$, $\begin{bmatrix}143&148\\166&33\end{bmatrix}$, $\begin{bmatrix}167&64\\148&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.cb.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 $ | $=$ | $ 24 y^{2} + z w $ |
| $=$ | $6 x^{3} + y z^{2} + y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 72 x^{5} + 2 x z^{4} + 3 y^{3} z^{2} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:1)$, $(0:0:1:0)$ |
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.cb.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 72X^{5}+3Y^{3}Z^{2}+2XZ^{4} $ |
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.k.1.8 | $168$ | $3$ | $3$ | $0$ | $?$ |
| 168.72.2-12.a.1.19 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.