Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $12^{4}\cdot24^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 9$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24C9 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}7&40\\108&119\end{bmatrix}$, $\begin{bmatrix}11&104\\84&67\end{bmatrix}$, $\begin{bmatrix}61&34\\24&95\end{bmatrix}$, $\begin{bmatrix}75&14\\32&113\end{bmatrix}$, $\begin{bmatrix}79&44\\104&105\end{bmatrix}$, $\begin{bmatrix}111&64\\92&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.er.1 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 $ | $=$ | $ t u - r s $ |
| $=$ | $z r - w s$ | |
| $=$ | $x t + y v$ | |
| $=$ | $x v - z s + w r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 162 x^{10} y^{2} - x^{8} y^{4} + 9 z^{12} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
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.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
| $\displaystyle W$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}+Z^{2}-W^{2} $ |
| $=$ | $ 8Y^{3}-XZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.9.er.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 9w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 162X^{10}Y^{2}-X^{8}Y^{4}+9Z^{12} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.96.1-24.bb.1.8 | $120$ | $3$ | $3$ | $1$ | $?$ |
| 120.144.4-24.c.1.14 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.c.1.27 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.cd.1.10 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.4-24.cd.1.40 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.144.5-24.h.1.9 | $120$ | $2$ | $2$ | $5$ | $?$ |
| 120.144.5-24.h.1.36 | $120$ | $2$ | $2$ | $5$ | $?$ |