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 | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.48.0.23 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&28\\6&31\end{bmatrix}$, $\begin{bmatrix}11&38\\24&29\end{bmatrix}$, $\begin{bmatrix}17&10\\18&31\end{bmatrix}$, $\begin{bmatrix}19&28\\24&49\end{bmatrix}$, $\begin{bmatrix}19&34\\48&19\end{bmatrix}$, $\begin{bmatrix}37&42\\24&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.24.0.a.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 110 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{24}(x^{2}+12y^{2})^{3}(x^{6}-60x^{4}y^{2}+1200x^{2}y^{4}+192y^{6})^{3}}{y^{6}x^{26}(x-6y)^{2}(x-2y)^{6}(x+2y)^{6}(x+6y)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 60.24.0-6.a.1.7 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-6.a.1.9 | $60$ | $2$ | $2$ | $0$ | $0$ |
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.a.1.4 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-12.a.1.9 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-12.a.1.15 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-12.a.2.1 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-12.a.2.12 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-12.a.2.15 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.a.1.1 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.a.1.28 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.a.1.29 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.a.2.5 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.a.2.20 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.a.2.25 | $60$ | $2$ | $2$ | $0$ |
| 60.96.1-12.a.1.3 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.a.1.11 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.a.1.10 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.a.1.11 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.b.1.4 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.b.1.12 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.b.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.b.1.18 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.c.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.c.1.9 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.c.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.c.1.7 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.d.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-12.d.1.8 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.d.1.2 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.d.1.20 | $60$ | $2$ | $2$ | $1$ |
| 60.96.2-12.a.1.2 | $60$ | $2$ | $2$ | $2$ |
| 60.96.2-12.a.2.4 | $60$ | $2$ | $2$ | $2$ |
| 60.96.2-60.a.1.2 | $60$ | $2$ | $2$ | $2$ |
| 60.96.2-60.a.2.6 | $60$ | $2$ | $2$ | $2$ |
| 60.144.1-6.a.1.4 | $60$ | $3$ | $3$ | $1$ |
| 60.240.8-30.a.1.10 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-30.a.1.19 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-30.a.1.20 | $60$ | $10$ | $10$ | $15$ |
| 120.96.0-24.o.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.1.19 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.o.2.21 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.1.56 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.1.57 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.40 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.49 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-24.bw.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bw.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bx.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bx.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.by.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.by.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bz.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bz.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dg.1.24 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dg.1.25 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dh.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dh.1.34 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.di.1.20 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.di.1.21 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dj.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dj.1.38 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-24.b.1.3 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-24.b.2.3 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.b.1.4 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.b.2.12 | $120$ | $2$ | $2$ | $2$ |
| 180.144.1-18.a.1.8 | $180$ | $3$ | $3$ | $1$ |
| 180.144.4-18.a.1.8 | $180$ | $3$ | $3$ | $4$ |
| 180.144.4-18.b.1.8 | $180$ | $3$ | $3$ | $4$ |