Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $24$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{6}\cdot6^{4}\cdot8^{2}\cdot12^{6}\cdot24^{2}$ | Cusp orbits | $2^{10}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AB5 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}17&48\\2&79\end{bmatrix}$, $\begin{bmatrix}17&72\\90&13\end{bmatrix}$, $\begin{bmatrix}71&72\\10&47\end{bmatrix}$, $\begin{bmatrix}95&96\\32&79\end{bmatrix}$, $\begin{bmatrix}97&12\\42&89\end{bmatrix}$, $\begin{bmatrix}97&24\\34&67\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.ci.2 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 $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 3 x w - 2 z^{2} - 2 z t $ |
| $=$ | $x w + 4 y^{2} - 2 z t - w^{2}$ | |
| $=$ | $3 x^{2} - 4 z t - 3 w^{2} + 2 t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 24 x^{6} z^{2} + 36 x^{4} y^{4} + 8 x^{4} z^{4} - 12 x^{2} y^{4} z^{2} - 16 x^{2} y^{2} z^{4} + \cdots + y^{4} z^{4} $ |
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 24.96.3.bm.3 :
| $\displaystyle X$ | $=$ | $\displaystyle 3y-z-t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3y+z+t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2z+t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y+2X^{2}Y^{2}+XY^{3}+2X^{2}YZ-2XY^{2}Z+2XYZ^{2}+XZ^{3}-YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.ci.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2z$ |
Equation of the image curve:
| $0$ | $=$ | $ 24X^{6}Z^{2}+36X^{4}Y^{4}+8X^{4}Z^{4}-12X^{2}Y^{4}Z^{2}-16X^{2}Y^{2}Z^{4}-2X^{2}Z^{6}+Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.192.1-24.ck.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ |
| 120.192.1-24.ck.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ |
| 120.192.3-24.bm.3.2 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.bm.3.39 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.bq.2.8 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.bq.2.27 | $120$ | $2$ | $2$ | $3$ | $?$ |