Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $72$ | ||
| 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 | $6^{2}\cdot12^{3}\cdot24$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24G4 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}25&20\\104&35\end{bmatrix}$, $\begin{bmatrix}77&74\\68&33\end{bmatrix}$, $\begin{bmatrix}83&50\\140&99\end{bmatrix}$, $\begin{bmatrix}93&34\\112&29\end{bmatrix}$, $\begin{bmatrix}95&82\\40&73\end{bmatrix}$, $\begin{bmatrix}111&94\\140&45\end{bmatrix}$, $\begin{bmatrix}111&122\\52&129\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.y.2 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $1032192$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 6 x y - z w $ |
| $=$ | $3 x^{3} + x z^{2} - 24 y^{3} - y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 6 x^{6} - x^{4} z^{2} + 18 x^{2} y^{3} z + 6 y^{3} z^{3} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2880x^{2}z^{10}-2592x^{2}z^{7}w^{3}-5388x^{2}z^{4}w^{6}-2307x^{2}zw^{9}-14592y^{2}z^{9}w-19776y^{2}z^{6}w^{4}-27648y^{2}z^{3}w^{7}-1530y^{2}w^{10}-64z^{12}-512z^{9}w^{3}-3060z^{6}w^{6}-2431z^{3}w^{9}-64w^{12}}{w^{3}z^{6}(3x^{2}z+18y^{2}w+z^{3}+w^{3})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.y.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -6X^{6}-X^{4}Z^{2}+18X^{2}Y^{3}Z+6Y^{3}Z^{3} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 84.72.2-12.b.1.9 | $84$ | $2$ | $2$ | $2$ | $?$ |
| 168.48.0-24.i.2.1 | $168$ | $3$ | $3$ | $0$ | $?$ |
| 168.72.2-12.b.1.16 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.