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: | 60.24.0.76 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&28\\2&47\end{bmatrix}$, $\begin{bmatrix}11&38\\38&21\end{bmatrix}$, $\begin{bmatrix}13&10\\36&41\end{bmatrix}$, $\begin{bmatrix}47&36\\52&49\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.12.0.b.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| 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 631 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot5^2}\cdot\frac{x^{12}(25x^{4}-1280x^{2}y^{2}+65536y^{4})^{3}}{y^{4}x^{16}(5x^{2}-256y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0-2.a.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 60.12.0-2.a.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 60.48.0-20.b.1.2 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-20.c.1.2 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-20.c.1.4 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.e.1.3 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.e.1.6 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.g.1.4 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.g.1.5 | $60$ | $2$ | $2$ | $0$ |
| 60.72.2-60.b.1.4 | $60$ | $3$ | $3$ | $2$ |
| 60.96.1-60.b.1.12 | $60$ | $4$ | $4$ | $1$ |
| 60.120.4-20.d.1.4 | $60$ | $5$ | $5$ | $4$ |
| 60.144.3-20.d.1.7 | $60$ | $6$ | $6$ | $3$ |
| 60.240.7-20.d.1.8 | $60$ | $10$ | $10$ | $7$ |
| 120.48.0-40.d.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.d.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.g.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.g.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.l.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.l.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.r.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.r.1.15 | $120$ | $2$ | $2$ | $0$ |