Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}\cdot12^{4}\cdot24^{4}$ | Cusp orbits | $2^{8}$ | ||
| 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: | 24AF9 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}11&42\\72&77\end{bmatrix}$, $\begin{bmatrix}45&106\\104&113\end{bmatrix}$, $\begin{bmatrix}57&98\\20&117\end{bmatrix}$, $\begin{bmatrix}61&90\\44&83\end{bmatrix}$, $\begin{bmatrix}71&14\\16&105\end{bmatrix}$, $\begin{bmatrix}109&46\\56&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.192.9.ct.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $12$ |
| Cyclic 120-torsion field degree: | $384$ |
| Full 120-torsion field degree: | $92160$ |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x t - w r $ |
| $=$ | $z t + w u + w v$ | |
| $=$ | $x y + x t - z s$ | |
| $=$ | $y v + t u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 162 x^{12} - 81 x^{8} y^{2} z^{2} - 36 x^{8} z^{4} + 18 x^{4} y^{2} z^{6} + 2 x^{4} z^{8} + \cdots - y^{2} z^{10} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=5,53$, and therefore no 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 24.96.5.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y-t$ |
| $\displaystyle W$ | $=$ | $\displaystyle -u$ |
| $\displaystyle T$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ YW-ZW+YT $ |
| $=$ | $ YZ-Z^{2}-2W^{2}+WT+T^{2} $ | |
| $=$ | $ 6X^{2}-ZW-YT $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.9.ct.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}s$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 162X^{12}-81X^{8}Y^{2}Z^{2}-36X^{8}Z^{4}+18X^{4}Y^{2}Z^{6}+2X^{4}Z^{8}-18Y^{4}Z^{8}-Y^{2}Z^{10} $ |
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 | $120$ | $4$ | $4$ | $1$ | $?$ |
| 120.192.3-24.f.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.f.1.29 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.ch.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.ch.1.34 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.5-24.h.1.13 | $120$ | $2$ | $2$ | $5$ | $?$ |
| 120.192.5-24.h.1.17 | $120$ | $2$ | $2$ | $5$ | $?$ |