Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $288$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $12^{4}\cdot24^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D9 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}17&80\\8&33\end{bmatrix}$, $\begin{bmatrix}31&10\\76&23\end{bmatrix}$, $\begin{bmatrix}61&4\\4&15\end{bmatrix}$, $\begin{bmatrix}75&34\\68&15\end{bmatrix}$, $\begin{bmatrix}85&94\\76&105\end{bmatrix}$, $\begin{bmatrix}117&100\\8&57\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.dn.2 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $1536$ |
| Full 120-torsion field degree: | $122880$ |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ u s + v r $ |
| $=$ | $x v + t s$ | |
| $=$ | $x u - t r$ | |
| $=$ | $y s + z v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{6} y^{2} - x^{6} z^{2} - 8 y^{6} z^{2} + y^{4} z^{4} $ |
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 |
|---|
| $(0:0:0:-1/2:0:1:0:0:0)$, $(0:0:0:1/2:0:1:0:0:0)$, $(0:0:0:1/2:0:0:1:0:0)$, $(0:0:0:-1/2:0:0:1:0:0)$ |
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 24.72.4.z.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle v$ |
| $\displaystyle W$ | $=$ | $\displaystyle r$ |
Equation of the image curve:
| $0$ | $=$ | $ 4XY-ZW $ |
| $=$ | $ 2X^{3}-16Y^{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 24.144.9.dn.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{6}Y^{2}-X^{6}Z^{2}-8Y^{6}Z^{2}+Y^{4}Z^{4} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
| 40.96.1-8.k.2.4 | $40$ | $3$ | $3$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.96.1-8.k.2.4 | $40$ | $3$ | $3$ | $1$ | $0$ |
| 120.144.4-24.z.2.2 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.z.2.53 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.5-24.c.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ |
| 120.144.5-24.c.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ |