Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $4$ are rational) | Cusp widths | $12^{8}\cdot24^{2}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 8$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24A8 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}7&4\\16&57\end{bmatrix}$, $\begin{bmatrix}31&144\\48&103\end{bmatrix}$, $\begin{bmatrix}47&32\\104&55\end{bmatrix}$, $\begin{bmatrix}103&28\\28&5\end{bmatrix}$, $\begin{bmatrix}137&52\\36&73\end{bmatrix}$, $\begin{bmatrix}145&72\\32&65\end{bmatrix}$, $\begin{bmatrix}151&92\\60&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.144.8.l.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $516096$ |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations
| $ 0 $ | $=$ | $ t v - u r $ |
| $=$ | $z w + t u$ | |
| $=$ | $x z - z^{2} + u^{2}$ | |
| $=$ | $x z + z^{2} - t^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{6} y^{2} - 2 y^{6} z^{2} + 5 y^{4} z^{4} - 4 y^{2} z^{6} + z^{8} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:-1:1:1:1:0:0)$, $(0:0:1:-1:1:1:0:0)$, $(0:0:1:1:-1:1:0:0)$, $(0:0:-1:-1:-1:1:0:0)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 12.72.4.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x$ |
| $\displaystyle W$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}-Y^{2}+Z^{2} $ |
| $=$ | $ XYZ+4W^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.8.l.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{6}Y^{2}-2Y^{6}Z^{2}+5Y^{4}Z^{4}-4Y^{2}Z^{6}+Z^{8} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
| 56.96.0-8.c.1.5 | $56$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.96.0-8.c.1.5 | $56$ | $3$ | $3$ | $0$ | $0$ |
| 168.144.4-12.b.1.9 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-12.b.1.29 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-24.z.1.11 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-24.z.1.54 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-24.z.2.33 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-24.z.2.64 | $168$ | $2$ | $2$ | $4$ | $?$ |