Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | ||||
| 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 | $1^{2}\cdot3^{2}\cdot4\cdot12$ | 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: | 12E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.48.0.220 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}8&13\\27&16\end{bmatrix}$, $\begin{bmatrix}25&44\\18&17\end{bmatrix}$, $\begin{bmatrix}40&21\\57&38\end{bmatrix}$, $\begin{bmatrix}41&14\\12&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.24.0.q.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 88 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot5^6}\cdot\frac{x^{24}(15x^{2}-64y^{2})^{3}(375x^{6}-4800x^{4}y^{2}+184320x^{2}y^{4}-262144y^{6})^{3}}{y^{4}x^{36}(5x^{2}-64y^{2})^{3}(45x^{2}-64y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-6.a.1.3 | $60$ | $2$ | $2$ | $0$ | $0$ |
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.b.1.9 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.h.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.r.1.4 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.t.1.7 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bl.1.2 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bn.1.4 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bo.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.br.1.12 | $60$ | $2$ | $2$ | $1$ |
| 60.144.1-60.s.1.7 | $60$ | $3$ | $3$ | $1$ |
| 60.240.8-60.bk.1.9 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-60.lx.1.20 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-60.fc.1.5 | $60$ | $10$ | $10$ | $15$ |
| 120.96.1-120.gg.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.kg.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bag.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bam.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byz.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzf.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzj.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzs.1.6 | $120$ | $2$ | $2$ | $1$ |
| 180.144.1-180.f.1.9 | $180$ | $3$ | $3$ | $1$ |
| 180.144.4-180.g.1.8 | $180$ | $3$ | $3$ | $4$ |
| 180.144.4-180.o.1.4 | $180$ | $3$ | $3$ | $4$ |