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: | $6$ (of which $2$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| 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: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.48.0.157 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}15&11\\28&37\end{bmatrix}$, $\begin{bmatrix}37&29\\44&39\end{bmatrix}$, $\begin{bmatrix}51&53\\8&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.24.0.d.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $46080$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 19 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^{10}\cdot5^2}\cdot\frac{x^{24}(25x^{4}-2560x^{2}y^{2}+16384y^{4})^{3}(25x^{4}+2560x^{2}y^{2}+16384y^{4})^{3}}{y^{4}x^{28}(25x^{4}+16384y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-4.d.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-4.d.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 |
|---|---|---|---|---|
| 60.240.8-20.g.1.1 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-20.q.1.2 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-20.q.1.3 | $60$ | $10$ | $10$ | $15$ |
| 120.96.1-40.cv.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.cx.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.di.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.dl.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.dx.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.ec.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.em.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.eo.1.3 | $120$ | $2$ | $2$ | $1$ |
| 60.144.4-60.bk.1.6 | $60$ | $3$ | $3$ | $4$ |
| 60.192.3-60.bh.1.2 | $60$ | $4$ | $4$ | $3$ |
| 120.96.1-120.jg.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.jk.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.le.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ll.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.mj.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ms.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.oa.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.oe.1.5 | $120$ | $2$ | $2$ | $1$ |