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Properties

Label 60.12.0.g.1
Level $60$
Index $12$
Genus $0$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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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: $4$ (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: 60.12.0.9

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}9&20\\32&33\end{bmatrix}$, $\begin{bmatrix}41&28\\9&31\end{bmatrix}$, $\begin{bmatrix}51&56\\14&37\end{bmatrix}$, $\begin{bmatrix}55&48\\18&43\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.24.0-60.g.1.1, 60.24.0-60.g.1.2, 60.24.0-60.g.1.3, 60.24.0-60.g.1.4, 120.24.0-60.g.1.1, 120.24.0-60.g.1.2, 120.24.0-60.g.1.3, 120.24.0-60.g.1.4, 120.24.0-60.g.1.5, 120.24.0-60.g.1.6, 120.24.0-60.g.1.7, 120.24.0-60.g.1.8, 120.24.0-60.g.1.9, 120.24.0-60.g.1.10, 120.24.0-60.g.1.11, 120.24.0-60.g.1.12
Cyclic 60-isogeny field degree: $24$
Cyclic 60-torsion field degree: $384$
Full 60-torsion field degree: $184320$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points, including 595 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 12 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3\cdot5}\cdot\frac{x^{12}(x^{4}-208x^{3}y-816x^{2}y^{2}+128xy^{3}+256y^{4})^{3}}{x^{14}(x+8y)^{2}(x^{2}+xy+4y^{2})^{4}}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_0(4)$ $4$ $2$ $2$ $0$ $0$
30.6.0.a.1 $30$ $2$ $2$ $0$ $0$
60.6.0.d.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.w.1 $60$ $3$ $3$ $2$
60.48.1.o.1 $60$ $4$ $4$ $1$
60.60.4.o.1 $60$ $5$ $5$ $4$
60.72.3.fe.1 $60$ $6$ $6$ $3$
60.120.7.w.1 $60$ $10$ $10$ $7$
120.24.0.dc.1 $120$ $2$ $2$ $0$
120.24.0.dd.1 $120$ $2$ $2$ $0$
120.24.0.do.1 $120$ $2$ $2$ $0$
120.24.0.dp.1 $120$ $2$ $2$ $0$
120.24.0.ds.1 $120$ $2$ $2$ $0$
120.24.0.dt.1 $120$ $2$ $2$ $0$
120.24.0.dw.1 $120$ $2$ $2$ $0$
120.24.0.dx.1 $120$ $2$ $2$ $0$
180.324.22.be.1 $180$ $27$ $27$ $22$