Invariants
| Level: | $60$ | $\SL_2$-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 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.48.0.104 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&22\\4&55\end{bmatrix}$, $\begin{bmatrix}9&28\\50&17\end{bmatrix}$, $\begin{bmatrix}17&20\\6&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.24.0.f.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $46080$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 5 x^{2} - 5 x y + 5 y^{2} + 12 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 |
|---|---|---|---|---|---|
| 12.24.0-12.b.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 20.24.0-20.a.1.2 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-20.a.1.3 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-12.b.1.3 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-60.b.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-60.b.1.6 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 60.144.4-60.i.1.5 | $60$ | $3$ | $3$ | $4$ |
| 60.192.3-60.i.1.5 | $60$ | $4$ | $4$ | $3$ |
| 60.240.8-60.i.1.3 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-60.bd.1.1 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-60.i.1.7 | $60$ | $10$ | $10$ | $15$ |