Modular curve 60.144.9.bx.2

• Label: 60.144.9.bx.2
• Level: $60$
• Index: $144$
• Genus: $9$
• Analytic rank: $0$
• 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	$4^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-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:	60.144.9.437

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}18&5\\19&33\end{bmatrix}$, $\begin{bmatrix}22&35\\35&11\end{bmatrix}$, $\begin{bmatrix}34&15\\51&19\end{bmatrix}$, $\begin{bmatrix}44&35\\55&47\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	60.288.9-60.bx.2.1, 60.288.9-60.bx.2.2, 60.288.9-60.bx.2.3, 60.288.9-60.bx.2.4, 120.288.9-60.bx.2.1, 120.288.9-60.bx.2.2, 120.288.9-60.bx.2.3, 120.288.9-60.bx.2.4
Cyclic 60-isogeny field degree:	$24$
Cyclic 60-torsion field degree:	$384$
Full 60-torsion field degree:	$15360$

Jacobian

Conductor:	$2^{24}\cdot3^{11}\cdot5^{9}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{5}\cdot2^{2}$
Newforms:	20.2.a.a, 36.2.a.a$^{2}$, 75.2.a.b, 100.2.a.a, 240.2.f.b, 720.2.f.d

Models

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

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

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{16} - 6 x^{14} z^{2} + 15 x^{12} y^{2} z^{2} + 7 x^{12} z^{4} + 10 x^{10} y^{2} z^{4} + \cdots + 961 z^{16} $

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=19$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$	$=$	$\displaystyle u$
$\displaystyle Y$	$=$	$\displaystyle s$
$\displaystyle Z$	$=$	$\displaystyle v$

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 30.72.5.f.1 :

$\displaystyle X$	$=$	$\displaystyle z$
$\displaystyle Y$	$=$	$\displaystyle t$
$\displaystyle Z$	$=$	$\displaystyle y$
$\displaystyle W$	$=$	$\displaystyle -u$
$\displaystyle T$	$=$	$\displaystyle -s$

Equation of the image curve:

$0$	$=$	$ X^{2}+YZ $
	$=$	$ 5X^{2}-5XY+25XZ-5YZ+W^{2} $
	$=$	$ X^{2}-3XY-Y^{2}+12XZ-31Z^{2}+XT-YT-ZT+T^{2} $

Modular covers

Hi

Sorry, your browser does not support the nearby lattice.

Cover information Click on a modular curve in the diagram to see information about it.

See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
30.72.5.f.1	$30$	$2$	$2$	$5$	$0$	$2^{2}$
60.72.1.fh.2	$60$	$2$	$2$	$1$	$0$	$1^{4}\cdot2^{2}$
60.72.3.zl.2	$60$	$2$	$2$	$3$	$0$	$1^{4}\cdot2$
60.72.3.bcp.2	$60$	$2$	$2$	$3$	$0$	$1^{4}\cdot2$
60.72.5.a.2	$60$	$2$	$2$	$5$	$0$	$1^{2}\cdot2$
60.72.5.s.2	$60$	$2$	$2$	$5$	$0$	$1^{2}\cdot2$
60.72.5.dx.2	$60$	$2$	$2$	$5$	$0$	$1^{4}$

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.v.1	$60$	$3$	$3$	$25$	$0$	$1^{8}\cdot2^{4}$
60.576.41.ek.1	$60$	$4$	$4$	$41$	$6$	$1^{16}\cdot2^{8}$
60.720.49.pj.1	$60$	$5$	$5$	$49$	$5$	$1^{18}\cdot2^{11}$