Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $192$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot6^{4}\cdot8^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $1^{2}\cdot2^{9}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 7$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 7$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AK7 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}1&72\\68&77\end{bmatrix}$, $\begin{bmatrix}25&96\\34&97\end{bmatrix}$, $\begin{bmatrix}29&12\\118&5\end{bmatrix}$, $\begin{bmatrix}55&48\\98&25\end{bmatrix}$, $\begin{bmatrix}67&60\\90&97\end{bmatrix}$, $\begin{bmatrix}79&24\\62&95\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.192.7.bn.4 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $12$ |
| Cyclic 120-torsion field degree: | $192$ |
| Full 120-torsion field degree: | $92160$ |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t - y z $ |
| $=$ | $x u - z v - w u$ | |
| $=$ | $2 z^{2} + t u$ | |
| $=$ | $x y + x v + y w - z t + z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 6 x^{6} y^{2} + 3 x^{6} z^{2} + 8 x^{4} y^{4} + 2 x^{4} y^{2} z^{2} + x^{4} z^{4} + 8 x^{2} y^{6} + \cdots + 4 y^{4} z^{4} $ |
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 |
|---|
| $(0:1:0:0:1:0:0)$, $(0:-1:0:0:1:0:0)$ |
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 24.96.3.bm.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
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 24.192.7.bn.4 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ -6X^{6}Y^{2}+3X^{6}Z^{2}+8X^{4}Y^{4}+2X^{4}Y^{2}Z^{2}+X^{4}Z^{4}+8X^{2}Y^{6}-12X^{2}Y^{4}Z^{2}+4X^{2}Y^{2}Z^{4}-8Y^{6}Z^{2}+4Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.192.3-24.bl.2.2 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.bl.2.23 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.bm.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.bm.1.45 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.bq.2.8 | $120$ | $2$ | $2$ | $3$ | $?$ |
| 120.192.3-24.bq.2.51 | $120$ | $2$ | $2$ | $3$ | $?$ |