Invariants
| Level: | $120$ | $\SL_2$-level: | $60$ | Newform level: | $240$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $13 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{2}\cdot12^{2}\cdot20^{2}\cdot60^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 13$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 60K13 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}11&33\\54&65\end{bmatrix}$, $\begin{bmatrix}11&57\\17&76\end{bmatrix}$, $\begin{bmatrix}100&113\\73&45\end{bmatrix}$, $\begin{bmatrix}101&96\\57&95\end{bmatrix}$, $\begin{bmatrix}103&98\\97&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.192.13.bh.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 $\mathbb{P}^{ 12 }$ defined by 55 equations
| $ 0 $ | $=$ | $ x y - x t + x u - x r + 2 x b + x d + y^{2} - 2 y w - y r - y a - y b - y c - 2 z w + z v - v^{2} + \cdots - c^{2} $ |
| $=$ | $2 x^{2} + 2 x u - 2 y^{2} - 2 y z + 4 y w - 2 z^{2} - z v + 2 v^{2} + v d + s a + a^{2}$ | |
| $=$ | $2 x^{2} - 2 x w - 3 x t + x v - 2 x r + x s + x a + 2 x b + y z + 2 y w - y t - y v - y r + 2 y a + \cdots - d^{2}$ | |
| $=$ | $x^{2} + 3 x y - x z + 2 x w + x t + x v + x r + x c + x d + y^{2} + y z + y w + 2 y t + y r - y s + \cdots + d^{2}$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
Map of degree 4 from the canonical model of this modular curve to the canonical model of the modular curve 20.48.3.j.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -z-w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y-z+w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y$ |
Equation of the image curve:
| $0$ | $=$ | $ 11X^{4}-4X^{3}Y+6X^{2}Y^{2}-4XY^{3}+11Y^{4}+22X^{3}Z+10X^{2}YZ-10XY^{2}Z-22Y^{3}Z+16X^{2}Z^{2}+36XYZ^{2}+16Y^{2}Z^{2}-4XZ^{3}+4YZ^{3}+2Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.96.1-60.bw.2.5 | $120$ | $4$ | $4$ | $1$ | $?$ |
| 120.192.7-60.g.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ |
| 120.192.7-60.g.1.14 | $120$ | $2$ | $2$ | $7$ | $?$ |