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: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.48.0.73 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}25&14\\54&41\end{bmatrix}$, $\begin{bmatrix}37&30\\12&19\end{bmatrix}$, $\begin{bmatrix}38&13\\3&32\end{bmatrix}$, $\begin{bmatrix}38&59\\3&46\end{bmatrix}$, $\begin{bmatrix}52&29\\45&56\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $6$ |
| Cyclic 60-torsion field degree: | $96$ |
| 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 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 60.24.0-6.a.1.4 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-6.a.1.12 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 60.96.0-12.c.1.3 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-12.c.2.4 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-12.c.3.7 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-12.c.4.8 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.c.1.9 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.c.2.9 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.c.3.9 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.c.4.9 | $60$ | $2$ | $2$ | $0$ |
| 60.96.1-12.b.1.11 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.h.1.7 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.k.1.3 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.k.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.l.1.2 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.l.1.9 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.o.1.8 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.p.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.144.1-12.f.1.3 | $60$ | $3$ | $3$ | $1$ |
| 60.240.8-60.o.1.15 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-60.gx.1.45 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-60.bq.1.46 | $60$ | $10$ | $10$ | $15$ |
| 120.96.0-24.bs.1.32 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.bs.2.30 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.bt.1.32 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.bt.2.30 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.bu.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.bu.2.12 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.bu.3.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.bu.4.14 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.dq.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.dq.2.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.dr.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.dr.2.14 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ds.1.29 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ds.2.27 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ds.3.57 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ds.4.53 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-24.cg.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.es.1.9 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.ik.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.in.1.9 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.iq.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.ir.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.is.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.it.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.iu.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.iv.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.iw.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.ix.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zc.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zf.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zo.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zr.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zu.1.29 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zv.1.29 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zw.1.27 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zx.1.27 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zy.1.27 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zz.1.27 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.baa.1.29 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bab.1.29 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-24.f.1.2 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-24.f.2.4 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-24.g.1.2 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-24.g.2.4 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.h.1.29 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.h.2.25 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.i.1.31 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.i.2.29 | $120$ | $2$ | $2$ | $2$ |
| 180.144.1-36.c.1.10 | $180$ | $3$ | $3$ | $1$ |
| 180.144.4-36.d.1.7 | $180$ | $3$ | $3$ | $4$ |
| 180.144.4-36.f.1.4 | $180$ | $3$ | $3$ | $4$ |