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 | $12^{8}\cdot24^{2}$ | 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: | 24A8 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}9&58\\64&91\end{bmatrix}$, $\begin{bmatrix}27&70\\4&91\end{bmatrix}$, $\begin{bmatrix}43&108\\48&55\end{bmatrix}$, $\begin{bmatrix}55&102\\28&35\end{bmatrix}$, $\begin{bmatrix}93&74\\32&59\end{bmatrix}$, $\begin{bmatrix}105&62\\8&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.144.8.bf.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 $\mathbb{P}^{ 7 }$ defined by 20 equations
| $ 0 $ | $=$ | $ 2 x v - t v - u r $ |
| $=$ | $2 x r - t v + t r + u v - u r$ | |
| $=$ | $x t + x u + z^{2} + z w - t u - u^{2}$ | |
| $=$ | $2 y u - w v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{6} y^{2} - 36 x^{6} y z + 9 x^{6} z^{2} + 2 y^{8} - 4 y^{7} z + 2 y^{6} z^{2} + 4 y^{5} z^{3} + \cdots + 2 z^{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 |
|---|
| $(0:0:-1:-1:1:1:0:0)$, $(0:0:1:1:1:1:0:0)$ |
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.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x-u$ |
| $\displaystyle W$ | $=$ | $\displaystyle -t$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{2}+Z^{2}-W^{2} $ |
| $=$ | $ 12Y^{3}-XZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.8.bf.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{6}Y^{2}-36X^{6}YZ+9X^{6}Z^{2}+2Y^{8}-4Y^{7}Z+2Y^{6}Z^{2}+4Y^{5}Z^{3}-8Y^{4}Z^{4}+4Y^{3}Z^{5}+2Y^{2}Z^{6}-4YZ^{7}+2Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.96.0-24.j.2.9 | $120$ | $3$ | $3$ | $0$ | $?$ |
| 120.144.4-12.e.1.27 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-12.e.1.30 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.y.1.17 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.y.1.37 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.z.2.25 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.z.2.55 | $120$ | $2$ | $2$ | $4$ | $?$ |