Modular curve 60.144.9.bv.1

• Label: 60.144.9.bv.1
• Level: $60$
• Index: $144$
• Genus: $9$
• Analytic rank: $2$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$60$		$\SL_2$-level:	$30$		Newform level:	$3600$
Index:	$144$		$\PSL_2$-index:	$144$
Genus:	$9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)		Cusp widths	$6^{4}\cdot30^{4}$		Cusp orbits	$2^{2}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$2$
$\Q$-gonality:	$4$
$\overline{\Q}$-gonality:	$4$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	30K9
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.144.9.151

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}6&25\\37&33\end{bmatrix}$, $\begin{bmatrix}9&25\\20&27\end{bmatrix}$, $\begin{bmatrix}38&55\\11&47\end{bmatrix}$, $\begin{bmatrix}54&5\\29&54\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	60.288.9-60.bv.1.1, 60.288.9-60.bv.1.2, 60.288.9-60.bv.1.3, 60.288.9-60.bv.1.4, 120.288.9-60.bv.1.1, 120.288.9-60.bv.1.2, 120.288.9-60.bv.1.3, 120.288.9-60.bv.1.4
Cyclic 60-isogeny field degree:	$24$
Cyclic 60-torsion field degree:	$384$
Full 60-torsion field degree:	$15360$

Jacobian

Conductor:	$2^{34}\cdot3^{14}\cdot5^{9}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{5}\cdot2^{2}$
Newforms:	80.2.a.b, 144.2.a.a$^{2}$, 240.2.f.b, 720.2.f.d, 900.2.a.b, 3600.2.a.u

Models

Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations

$ 0 $	$=$	$ z r - t u $
	$=$	$x r + w u$
	$=$	$x t + z w$
	$=$	$x w - y t$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 1875 x^{16} - 1625 x^{14} y^{2} - 4500 x^{14} z^{2} + 300 x^{12} y^{4} - 300 x^{12} y^{2} z^{2} + \cdots + 19683 z^{16} $

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 canonical model to plane model:

$\displaystyle X$	$=$	$\displaystyle u$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{5}s$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{3}r$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 60.72.5.o.1 :

$\displaystyle X$	$=$	$\displaystyle -x$
$\displaystyle Y$	$=$	$\displaystyle -y$
$\displaystyle Z$	$=$	$\displaystyle -z$
$\displaystyle W$	$=$	$\displaystyle w-t+2r$
$\displaystyle T$	$=$	$\displaystyle s$

Equation of the image curve:

$0$	$=$	$ X^{2}+YZ $
	$=$	$ 15X^{2}-15XY+75XZ-15YZ-W^{2} $
	$=$	$ 22X^{2}+15Y^{2}-23YZ+375Z^{2}-2W^{2}-T^{2} $

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
30.72.1.o.1	$30$	$2$	$2$	$1$	$1$	$1^{4}\cdot2^{2}$
60.72.3.zj.2	$60$	$2$	$2$	$3$	$1$	$1^{4}\cdot2$
60.72.3.bci.1	$60$	$2$	$2$	$3$	$1$	$1^{4}\cdot2$
60.72.5.f.1	$60$	$2$	$2$	$5$	$0$	$1^{2}\cdot2$
60.72.5.o.1	$60$	$2$	$2$	$5$	$2$	$2^{2}$
60.72.5.u.2	$60$	$2$	$2$	$5$	$0$	$1^{2}\cdot2$
60.72.5.eg.1	$60$	$2$	$2$	$5$	$1$	$1^{4}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.432.25.ch.2	$60$	$3$	$3$	$25$	$5$	$1^{8}\cdot2^{4}$
60.576.41.eh.1	$60$	$4$	$4$	$41$	$9$	$1^{16}\cdot2^{8}$
60.720.49.qc.1	$60$	$5$	$5$	$49$	$9$	$1^{18}\cdot2^{11}$