Invariants
| Level: | $60$ | $\SL_2$-level: | $4$ | ||||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $1^{2}\cdot4$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.6.0.2 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&14\\57&59\end{bmatrix}$, $\begin{bmatrix}27&2\\20&43\end{bmatrix}$, $\begin{bmatrix}31&36\\26&23\end{bmatrix}$, $\begin{bmatrix}35&14\\33&43\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $368640$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 2787 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2^{12}}\cdot\frac{(15x+4y)^{6}(5x^{2}-1024y^{2})^{3}}{y^{4}(15x+4y)^{6}(15x^{2}-4096y^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 60.12.0.a.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.c.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.r.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.s.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.ba.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.bb.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.bi.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.bj.1 | $60$ | $2$ | $2$ | $0$ |
| 60.18.1.e.1 | $60$ | $3$ | $3$ | $1$ |
| 60.24.0.s.1 | $60$ | $4$ | $4$ | $0$ |
| 60.30.2.f.1 | $60$ | $5$ | $5$ | $2$ |
| 60.36.1.bg.1 | $60$ | $6$ | $6$ | $1$ |
| 60.60.3.bf.1 | $60$ | $10$ | $10$ | $3$ |
| 120.12.0.c.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.h.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.ce.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.ch.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.de.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.dh.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.ec.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.ef.1 | $120$ | $2$ | $2$ | $0$ |
| 180.162.10.e.1 | $180$ | $27$ | $27$ | $10$ |