Invariants
| Level: | $60$ | $\SL_2$-level: | $6$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.16.0.38 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&17\\39&32\end{bmatrix}$, $\begin{bmatrix}16&47\\13&3\end{bmatrix}$, $\begin{bmatrix}54&47\\55&26\end{bmatrix}$, $\begin{bmatrix}56&31\\3&4\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.8.0.a.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $36$ |
| Cyclic 60-torsion field degree: | $576$ |
| Full 60-torsion field degree: | $138240$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 222 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6}\cdot\frac{x^{8}(x^{2}-108y^{2})(x^{2}-12y^{2})^{3}}{y^{6}x^{10}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.2.0.a.1 | $4$ | $8$ | $4$ | $0$ | $0$ |
| 15.8.0-3.a.1.2 | $15$ | $2$ | $2$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 15.8.0-3.a.1.2 | $15$ | $2$ | $2$ | $0$ | $0$ |
| 60.8.0-3.a.1.4 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 60.32.0-12.b.1.3 | $60$ | $2$ | $2$ | $0$ |
| 60.32.0-12.b.2.4 | $60$ | $2$ | $2$ | $0$ |
| 60.32.0-60.b.1.4 | $60$ | $2$ | $2$ | $0$ |
| 60.32.0-60.b.2.7 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-12.d.1.2 | $60$ | $3$ | $3$ | $0$ |
| 60.48.1-12.b.1.1 | $60$ | $3$ | $3$ | $1$ |
| 60.64.1-12.b.1.1 | $60$ | $4$ | $4$ | $1$ |
| 60.80.2-60.a.1.1 | $60$ | $5$ | $5$ | $2$ |
| 60.96.3-60.c.1.23 | $60$ | $6$ | $6$ | $3$ |
| 60.160.5-60.a.1.18 | $60$ | $10$ | $10$ | $5$ |
| 120.32.0-24.b.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.32.0-24.b.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.32.0-120.b.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.32.0-120.b.2.11 | $120$ | $2$ | $2$ | $0$ |
| 180.48.0-36.b.1.4 | $180$ | $3$ | $3$ | $0$ |
| 180.48.1-36.a.1.2 | $180$ | $3$ | $3$ | $1$ |
| 180.48.2-36.a.1.2 | $180$ | $3$ | $3$ | $2$ |