Invariants
Level: | $60$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
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: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.24.0.26 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}11&12\\45&13\end{bmatrix}$, $\begin{bmatrix}43&40\\47&39\end{bmatrix}$, $\begin{bmatrix}49&56\\30&43\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 60.12.0.h.1 for the level structure with $-I$) |
Cyclic 60-isogeny field degree: | $24$ |
Cyclic 60-torsion field degree: | $384$ |
Full 60-torsion field degree: | $92160$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 472 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3\cdot5}\cdot\frac{x^{12}(921600x^{4}+13440x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{14}(960x^{2}-y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
4.12.0-4.c.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ |
60.12.0-4.c.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
60.72.2-60.x.1.7 | $60$ | $3$ | $3$ | $2$ |
60.96.1-60.p.1.2 | $60$ | $4$ | $4$ | $1$ |
60.120.4-60.p.1.3 | $60$ | $5$ | $5$ | $4$ |
60.144.3-60.ff.1.7 | $60$ | $6$ | $6$ | $3$ |
60.240.7-60.x.1.1 | $60$ | $10$ | $10$ | $7$ |
120.48.0-120.dg.1.1 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dg.1.16 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dh.1.1 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dh.1.24 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dk.1.2 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dk.1.15 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dl.1.2 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dl.1.15 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ea.1.2 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ea.1.15 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.eb.1.2 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.eb.1.15 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ee.1.1 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ee.1.16 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ef.1.1 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ef.1.16 | $120$ | $2$ | $2$ | $0$ |