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: | 60.48.0.255 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}4&1\\51&52\end{bmatrix}$, $\begin{bmatrix}28&21\\33&38\end{bmatrix}$, $\begin{bmatrix}49&16\\24&59\end{bmatrix}$, $\begin{bmatrix}50&13\\9&38\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.24.0.b.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 80 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8\cdot11^3}{3^3\cdot5^3}\cdot\frac{(2x+9y)^{24}(x^{2}-xy-11y^{2})^{3}(109x^{6}-12567x^{5}y+199140x^{4}y^{2}-826135x^{3}y^{3}+2286060x^{2}y^{4}-3311247xy^{5}+3436201y^{6})^{3}}{(x-23y)^{2}(x-y)^{6}(2x+9y)^{24}(4x^{2}-19xy+136y^{2})^{6}(34x^{2}-79xy+166y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 60.16.0-30.b.1.4 | $60$ | $3$ | $3$ | $0$ | $0$ |
| 60.24.0-6.a.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ |
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.m.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.o.1.9 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bd.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.be.1.4 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bl.1.4 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bm.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bs.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.bu.1.10 | $60$ | $2$ | $2$ | $1$ |
| 60.144.1-30.e.1.2 | $60$ | $3$ | $3$ | $1$ |
| 60.240.8-30.h.1.7 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-30.d.1.7 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-30.p.1.16 | $60$ | $10$ | $10$ | $15$ |
| 120.96.1-120.zj.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zp.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.blm.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.blp.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bza.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzd.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzu.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.caa.1.6 | $120$ | $2$ | $2$ | $1$ |
| 180.144.1-90.e.1.7 | $180$ | $3$ | $3$ | $1$ |
| 180.144.4-90.g.1.4 | $180$ | $3$ | $3$ | $4$ |
| 180.144.4-90.j.1.4 | $180$ | $3$ | $3$ | $4$ |