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: | 60.48.0.217 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&26\\12&23\end{bmatrix}$, $\begin{bmatrix}16&33\\45&52\end{bmatrix}$, $\begin{bmatrix}19&24\\6&23\end{bmatrix}$, $\begin{bmatrix}31&36\\36&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.24.0.p.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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot5^6}\cdot\frac{x^{24}(15x^{2}+16y^{2})^{3}(375x^{6}+1200x^{4}y^{2}+11520x^{2}y^{4}+4096y^{6})^{3}}{y^{4}x^{36}(5x^{2}+16y^{2})^{3}(45x^{2}+16y^{2})}$ |
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 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 30.24.0-6.a.1.4 | $30$ | $2$ | $2$ | $0$ | $0$ |
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.a.1.13 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.e.1.10 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.q.1.4 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.s.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bk.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bm.1.8 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bp.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bq.1.12 | $60$ | $2$ | $2$ | $1$ |
| 60.144.1-60.r.1.5 | $60$ | $3$ | $3$ | $1$ |
| 60.240.8-60.bj.1.13 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-60.lw.1.18 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-60.fb.1.2 | $60$ | $10$ | $10$ | $15$ |
| 120.96.1-120.gf.1.9 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.jx.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bad.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.baj.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byw.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzc.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzm.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzp.1.5 | $120$ | $2$ | $2$ | $1$ |
| 180.144.1-180.e.1.11 | $180$ | $3$ | $3$ | $1$ |
| 180.144.4-180.f.1.6 | $180$ | $3$ | $3$ | $4$ |
| 180.144.4-180.n.1.8 | $180$ | $3$ | $3$ | $4$ |