Modular curve 60.120.5.x.1

• Label: 60.120.5.x.1
• Level: $60$
• Index: $120$
• Genus: $5$
• Analytic rank: $3$
• Cusps: $12$
• $\Q$-cusps: $0$

Invariants

Level:	$60$		$\SL_2$-level:	$10$		Newform level:	$3600$
Index:	$120$		$\PSL_2$-index:	$120$
Genus:	$5 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (none of which are rational)		Cusp widths	$10^{12}$		Cusp orbits	$4\cdot8$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$3$
$\Q$-gonality:	$2 \le \gamma \le 4$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	10A5
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.120.5.111

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}19&25\\50&49\end{bmatrix}$, $\begin{bmatrix}53&36\\43&7\end{bmatrix}$, $\begin{bmatrix}53&39\\40&7\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 60-isogeny field degree:	$48$
Cyclic 60-torsion field degree:	$768$
Full 60-torsion field degree:	$18432$

Jacobian

Conductor:	$2^{14}\cdot3^{10}\cdot5^{8}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{5}$
Newforms:	180.2.a.a, 720.2.a.h, 900.2.a.b$^{2}$, 3600.2.a.be

Models

Embedded model Embedded model in $\mathbb{P}^{7}$

$ 0 $	$=$	$ x^{2} - x y + x w - y w + t u - r^{2} $
	$=$	$x z + x w - y z - y w + t^{2} - t v$
	$=$	$2 x^{2} + x z - 2 y^{2} - y z - t v + r^{2}$
	$=$	$x u - x v + z v + 2 w t - w v$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 78125 x^{12} + 187500 x^{10} y^{2} - 187500 x^{10} z^{2} + 191250 x^{8} y^{4} - 345000 x^{8} y^{2} z^{2} + \cdots + 25 z^{12} $

Geometric Weierstrass model Geometric Weierstrass model

$ 2025 w^{2} $	$=$	$ 2375 x^{6} + 2750 x^{5} y - 625 x^{4} z^{2} + 3300 x^{3} y z^{2} + 945 x^{2} z^{4} - 2970 x y z^{4} - 279 z^{6} $
$0$	$=$	$2 x^{2} + 2 x y + 3 y^{2} - z^{2}$

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$	$=$	$\displaystyle w$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{3}r$
$\displaystyle Z$	$=$	$\displaystyle v$

Maps to other modular curves

$j$-invariant map of degree 120 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{3^2}{2^2}\cdot\frac{415800w^{2}v^{8}-1780200w^{2}v^{6}r^{2}+326700w^{2}v^{4}r^{4}-10800w^{2}v^{2}r^{6}-94743tuv^{8}+263547tuv^{6}r^{2}-3996tuv^{4}r^{4}-5940tuv^{2}r^{6}+576tur^{8}-45180tv^{7}r^{2}-8460tv^{5}r^{4}-1600tv^{3}r^{6}+18117u^{2}v^{8}-87849u^{2}v^{6}r^{2}+17856u^{2}v^{4}r^{4}-648u^{2}v^{2}r^{6}+18117uv^{9}-8649uv^{7}r^{2}-10224uv^{5}r^{4}-4768uv^{3}r^{6}-167103v^{10}+811935v^{8}r^{2}-436200v^{6}r^{4}+17660v^{4}r^{6}+7920v^{2}r^{8}-768r^{10}}{r^{10}}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
20.60.0.b.1	$20$	$2$	$2$	$0$	$0$	full Jacobian
30.60.3.d.1	$30$	$2$	$2$	$3$	$2$	$1^{2}$
60.40.1.g.1	$60$	$3$	$3$	$1$	$1$	$1^{4}$
60.40.1.j.1	$60$	$3$	$3$	$1$	$1$	$1^{4}$
60.60.2.g.1	$60$	$2$	$2$	$2$	$1$	$1^{3}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.360.13.v.1	$60$	$3$	$3$	$13$	$5$	$1^{8}$
60.360.25.lu.1	$60$	$3$	$3$	$25$	$13$	$1^{14}\cdot2^{3}$
60.360.25.lv.1	$60$	$3$	$3$	$25$	$11$	$1^{14}\cdot2^{3}$
60.480.29.hv.1	$60$	$4$	$4$	$29$	$9$	$1^{24}$
60.480.29.hz.1	$60$	$4$	$4$	$29$	$14$	$1^{24}$