Invariants
| Level: | $60$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.24.0.98 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&20\\6&59\end{bmatrix}$, $\begin{bmatrix}11&40\\49&7\end{bmatrix}$, $\begin{bmatrix}51&32\\14&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.12.0.h.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $92160$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 472 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3\cdot5}\cdot\frac{x^{12}(921600x^{4}+13440x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{14}(960x^{2}-y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0-4.c.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 20.12.0-4.c.1.2 | $20$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 60.72.2-60.x.1.5 | $60$ | $3$ | $3$ | $2$ |
| 60.96.1-60.p.1.3 | $60$ | $4$ | $4$ | $1$ |
| 60.120.4-60.p.1.4 | $60$ | $5$ | $5$ | $4$ |
| 60.144.3-60.ff.1.1 | $60$ | $6$ | $6$ | $3$ |
| 60.240.7-60.x.1.5 | $60$ | $10$ | $10$ | $7$ |
| 120.48.0-120.dg.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dg.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dh.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dh.1.17 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dk.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dk.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dl.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dl.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ea.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ea.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.eb.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.eb.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ee.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ee.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ef.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ef.1.9 | $120$ | $2$ | $2$ | $0$ |