Invariants
Level: | $60$ | $\SL_2$-level: | $4$ | ||||
Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.12.0.22 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&28\\26&9\end{bmatrix}$, $\begin{bmatrix}13&22\\12&17\end{bmatrix}$, $\begin{bmatrix}15&1\\28&9\end{bmatrix}$, $\begin{bmatrix}21&53\\58&1\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 60.24.0-60.d.1.1, 60.24.0-60.d.1.2, 60.24.0-60.d.1.3, 60.24.0-60.d.1.4, 120.24.0-60.d.1.1, 120.24.0-60.d.1.2, 120.24.0-60.d.1.3, 120.24.0-60.d.1.4 |
Cyclic 60-isogeny field degree: | $48$ |
Cyclic 60-torsion field degree: | $768$ |
Full 60-torsion field degree: | $184320$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 64 x^{2} + 15 y^{2} + 15 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
4.6.0.a.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
60.6.0.c.1 | $60$ | $2$ | $2$ | $0$ | $0$ |
60.6.0.e.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 |
---|---|---|---|---|
60.36.2.o.1 | $60$ | $3$ | $3$ | $2$ |
60.48.1.g.1 | $60$ | $4$ | $4$ | $1$ |
60.60.4.g.1 | $60$ | $5$ | $5$ | $4$ |
60.72.3.dh.1 | $60$ | $6$ | $6$ | $3$ |
60.120.7.o.1 | $60$ | $10$ | $10$ | $7$ |
180.324.22.s.1 | $180$ | $27$ | $27$ | $22$ |