Invariants
| Level: | $60$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $1^{2}\cdot4$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 4B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.12.0.7 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&44\\48&7\end{bmatrix}$, $\begin{bmatrix}33&28\\1&47\end{bmatrix}$, $\begin{bmatrix}37&16\\5&3\end{bmatrix}$, $\begin{bmatrix}57&32\\4&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.6.0.c.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $184320$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 95098 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(48x^{2}-y^{2})^{3}}{x^{10}(8x-y)(8x+y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 60.24.0-4.b.1.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-4.d.1.1 | $60$ | $2$ | $2$ | $0$ |
| 120.24.0-8.d.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-8.k.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-8.m.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-8.m.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-8.n.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-8.n.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-8.o.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-8.o.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-8.p.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-8.p.1.5 | $120$ | $2$ | $2$ | $0$ |
| 60.24.0-12.g.1.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-12.h.1.2 | $60$ | $2$ | $2$ | $0$ |
| 60.36.1-12.c.1.2 | $60$ | $3$ | $3$ | $1$ |
| 60.48.0-12.g.1.3 | $60$ | $4$ | $4$ | $0$ |
| 60.24.0-20.g.1.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-20.h.1.2 | $60$ | $2$ | $2$ | $0$ |
| 60.60.2-20.c.1.3 | $60$ | $5$ | $5$ | $2$ |
| 60.72.1-20.c.1.4 | $60$ | $6$ | $6$ | $1$ |
| 60.120.3-20.c.1.5 | $60$ | $10$ | $10$ | $3$ |
| 120.24.0-24.s.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.v.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.y.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.y.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.z.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.z.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.ba.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.ba.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.bb.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-24.bb.1.10 | $120$ | $2$ | $2$ | $0$ |
| 180.324.10-36.d.1.2 | $180$ | $27$ | $27$ | $10$ |
| 120.24.0-40.s.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.v.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.y.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.y.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.z.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.z.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.ba.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.ba.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.bb.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-40.bb.1.16 | $120$ | $2$ | $2$ | $0$ |
| 60.24.0-60.g.1.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-60.h.1.3 | $60$ | $2$ | $2$ | $0$ |
| 120.24.0-120.s.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.v.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.y.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.y.1.17 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.z.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.z.1.21 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.ba.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.ba.1.21 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.bb.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.bb.1.17 | $120$ | $2$ | $2$ | $0$ |