Invariants
| Level: | $120$ | $\SL_2$-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 $2$ are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24B8 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}31&62\\32&47\end{bmatrix}$, $\begin{bmatrix}39&20\\44&67\end{bmatrix}$, $\begin{bmatrix}65&108\\28&25\end{bmatrix}$, $\begin{bmatrix}73&50\\60&59\end{bmatrix}$, $\begin{bmatrix}89&72\\52&13\end{bmatrix}$, $\begin{bmatrix}97&70\\64&83\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.144.8.er.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $122880$ |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations
| $ 0 $ | $=$ | $ t^{2} - t u + u^{2} + v r $ |
| $=$ | $3 w^{2} + v r$ | |
| $=$ | $2 w t - w u - w r - u r - v r$ | |
| $=$ | $w v + w r - u v + u r + 2 v r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 3 x^{8} + 12 x^{7} y - 3 x^{6} y^{2} - 60 x^{5} y^{3} + 7 x^{5} z^{3} + 159 x^{4} y^{4} + \cdots + 24 y^{8} $ |
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 |
|---|
| $(0:0:0:0:0:0:1:0)$, $(0:0:0:0:0:0:0:1)$ |
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 24.72.4.y.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x-y-2z$ |
| $\displaystyle W$ | $=$ | $\displaystyle v+r$ |
Equation of the image curve:
| $0$ | $=$ | $ 6XY-ZW $ |
| $=$ | $ 3X^{3}-24Y^{3}+XZ^{2}-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.8.er.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3w$ |
Equation of the image curve:
| $0$ | $=$ | $ -3X^{8}+12X^{7}Y-3X^{6}Y^{2}-60X^{5}Y^{3}+7X^{5}Z^{3}+159X^{4}Y^{4}-20X^{4}YZ^{3}-222X^{3}Y^{5}+24X^{3}Y^{2}Z^{3}+186X^{2}Y^{6}-8X^{2}Y^{3}Z^{3}-96XY^{7}+4XY^{4}Z^{3}+24Y^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.144.4-24.y.2.37 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.y.2.54 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.z.1.5 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.z.1.24 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.cd.1.18 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.cd.1.31 | $120$ | $2$ | $2$ | $4$ | $?$ |