Invariants
| Level: | $60$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $4^{3}$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $2$ 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: | 4F0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.12.0.16 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&22\\18&19\end{bmatrix}$, $\begin{bmatrix}25&46\\9&35\end{bmatrix}$, $\begin{bmatrix}29&22\\5&27\end{bmatrix}$, $\begin{bmatrix}37&18\\0&7\end{bmatrix}$ |
| 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: | $184320$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 23 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2^8\cdot5^2}\cdot\frac{(2x-y)^{12}(16x^{2}+56xy-11y^{2})^{3}(48x^{2}+8xy+7y^{2})^{3}}{(2x-y)^{12}(4x+y)^{4}(4x^{2}-xy+y^{2})^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 60.24.0.b.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.h.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.l.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.m.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.v.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.w.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.z.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0.bc.1 | $60$ | $2$ | $2$ | $0$ |
| 60.36.1.ft.1 | $60$ | $3$ | $3$ | $1$ |
| 60.48.2.f.1 | $60$ | $4$ | $4$ | $2$ |
| 60.60.4.cj.1 | $60$ | $5$ | $5$ | $4$ |
| 60.72.3.zt.1 | $60$ | $6$ | $6$ | $3$ |
| 60.120.7.jx.1 | $60$ | $10$ | $10$ | $7$ |
| 120.24.0.bd.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.bn.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.et.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ff.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gh.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.gp.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ht.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.il.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.jd.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.jf.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.lf.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ll.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.lp.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.lr.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ml.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mr.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mv.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mx.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nr.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nx.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.od.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.of.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.oj.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.op.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.bh.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.bn.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.br.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.bt.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.fx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gd.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gz.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.kj.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.kp.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.lj.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ll.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.lp.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.lv.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.nv.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.nx.1 | $120$ | $2$ | $2$ | $1$ |
| 180.324.22.hx.1 | $180$ | $27$ | $27$ | $22$ |