Invariants
Level: | $120$ | $\SL_2$-level: | $4$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4G0 |
Level structure
$\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}11&8\\102&59\end{bmatrix}$, $\begin{bmatrix}19&4\\74&103\end{bmatrix}$, $\begin{bmatrix}41&102\\14&67\end{bmatrix}$, $\begin{bmatrix}51&110\\2&61\end{bmatrix}$, $\begin{bmatrix}103&36\\88&95\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 60.24.0.b.1 for the level structure with $-I$) |
Cyclic 120-isogeny field degree: | $96$ |
Cyclic 120-torsion field degree: | $3072$ |
Full 120-torsion field degree: | $737280$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ x^{2} - x y + 4 y^{2} + 15 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.24.0-4.a.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-4.a.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
120.24.0-60.b.1.1 | $120$ | $2$ | $2$ | $0$ | $?$ |
120.24.0-60.b.1.4 | $120$ | $2$ | $2$ | $0$ | $?$ |
120.24.0-60.b.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
120.96.1-120.j.1.5 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.k.1.8 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.cr.1.7 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.cs.1.7 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.er.1.8 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.es.1.7 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.fl.1.7 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.fm.1.5 | $120$ | $2$ | $2$ | $1$ |
120.144.4-60.e.1.5 | $120$ | $3$ | $3$ | $4$ |
120.192.3-60.e.1.4 | $120$ | $4$ | $4$ | $3$ |
120.240.8-60.e.1.4 | $120$ | $5$ | $5$ | $8$ |
120.288.7-60.z.1.4 | $120$ | $6$ | $6$ | $7$ |
120.480.15-60.e.1.19 | $120$ | $10$ | $10$ | $15$ |