Invariants
| Level: | $60$ | $\SL_2$-level: | $6$ | ||||
| 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 | $2^{3}\cdot6^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 6I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.48.0.52 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}17&24\\42&43\end{bmatrix}$, $\begin{bmatrix}29&16\\24&37\end{bmatrix}$, $\begin{bmatrix}44&43\\27&50\end{bmatrix}$, $\begin{bmatrix}56&45\\3&2\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.24.0.o.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| 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 60 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^6\cdot5^3}\cdot\frac{(2x-5y)^{24}(28x^{2}+40xy-5y^{2})^{3}(12608x^{6}-59520x^{5}y-27600x^{4}y^{2}+56000x^{3}y^{3}+97500x^{2}y^{4}+51000xy^{5}+13625y^{6})^{3}}{(2x-5y)^{26}(2x+y)^{6}(4x^{2}+5y^{2})^{6}(92x^{2}+80xy+35y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-6.a.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 60.16.0-60.a.1.2 | $60$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 60.96.1-60.j.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.l.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.v.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.x.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bh.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bj.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bp.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.br.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.144.1-60.o.1.2 | $60$ | $3$ | $3$ | $1$ |
| 60.240.8-60.bi.1.6 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-60.lv.1.3 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-60.fa.1.26 | $60$ | $10$ | $10$ | $15$ |
| 120.96.1-120.za.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zg.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bas.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bay.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byn.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byt.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzl.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzr.1.1 | $120$ | $2$ | $2$ | $1$ |
| 180.144.1-180.d.1.8 | $180$ | $3$ | $3$ | $1$ |
| 180.144.4-180.e.1.5 | $180$ | $3$ | $3$ | $4$ |
| 180.144.4-180.m.1.14 | $180$ | $3$ | $3$ | $4$ |