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: | 60.48.0.53 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}2&37\\27&8\end{bmatrix}$, $\begin{bmatrix}16&37\\45&8\end{bmatrix}$, $\begin{bmatrix}28&41\\45&26\end{bmatrix}$, $\begin{bmatrix}50&27\\27&56\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.24.0.r.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 58 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^3\cdot5^3}\cdot\frac{x^{24}(45x^{2}+4y^{2})^{3}(10125x^{6}+67500x^{4}y^{2}-3600x^{2}y^{4}+64y^{6})^{3}}{y^{2}x^{30}(15x^{2}-4y^{2})^{6}(135x^{2}-4y^{2})^{2}}$ |
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 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 60.16.0-60.b.1.3 | $60$ | $3$ | $3$ | $0$ | $0$ |
| 60.24.0-6.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 |
|---|---|---|---|---|
| 60.96.1-60.n.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.p.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bc.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bf.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bk.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bn.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bt.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bv.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.144.1-60.bg.1.2 | $60$ | $3$ | $3$ | $1$ |
| 60.240.8-60.bm.1.10 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-60.ly.1.1 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-60.fd.1.12 | $60$ | $10$ | $10$ | $15$ |
| 120.96.1-120.zm.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zs.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.blj.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bls.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byx.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzg.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzx.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.cad.1.1 | $120$ | $2$ | $2$ | $1$ |
| 180.144.1-180.g.1.8 | $180$ | $3$ | $3$ | $1$ |
| 180.144.4-180.i.1.5 | $180$ | $3$ | $3$ | $4$ |
| 180.144.4-180.p.1.13 | $180$ | $3$ | $3$ | $4$ |