Modular curve 60.144.5.kw.2

• Label: 60.144.5.kw.2
• Level: $60$
• Index: $144$
• Genus: $5$
• Analytic rank: $2$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$60$		$\SL_2$-level:	$20$		Newform level:	$3600$
Index:	$144$		$\PSL_2$-index:	$144$
Genus:	$5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)		Cusp widths	$2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$		Cusp orbits	$4^{4}$
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:	20I5
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.144.5.924

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}17&15\\2&1\end{bmatrix}$, $\begin{bmatrix}29&55\\6&19\end{bmatrix}$, $\begin{bmatrix}53&20\\50&27\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 60-isogeny field degree:	$8$
Cyclic 60-torsion field degree:	$128$
Full 60-torsion field degree:	$15360$

Jacobian

Conductor:	$2^{16}\cdot3^{4}\cdot5^{7}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{3}\cdot2$
Newforms:	40.2.c.a, 180.2.a.a, 400.2.a.e, 3600.2.a.be

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $	$=$	$ x^{2} + 3 x y + x z + z^{2} $
	$=$	$x^{2} + 3 x y - 4 x z + 5 y^{2} - 4 z^{2} + w^{2} + t^{2}$
	$=$	$3 x^{2} - 6 x y + 8 x z + 10 y^{2} + 8 z^{2} - w^{2} - 2 t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 13125 x^{8} + 1500 x^{7} y + 325 x^{6} y^{2} - 22500 x^{6} z^{2} + 10 x^{5} y^{3} - 3150 x^{5} y z^{2} + \cdots + 37746 z^{8} $

Rational points

This modular curve has no real points, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$	$=$	$\displaystyle x-y$
$\displaystyle Y$	$=$	$\displaystyle 5z+5w$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{3}t$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -3^3\,\frac{56949480xzw^{16}+265764240xzw^{14}t^{2}+392027040xzw^{12}t^{4}+23950080xzw^{10}t^{6}-495590400xzw^{8}t^{8}-492549120xzw^{6}t^{10}-175196160xzw^{4}t^{12}-15298560xzw^{2}t^{14}+1351680xzt^{16}+56949480z^{2}w^{16}+265764240z^{2}w^{14}t^{2}+392027040z^{2}w^{12}t^{4}+23950080z^{2}w^{10}t^{6}-495590400z^{2}w^{8}t^{8}-492549120z^{2}w^{6}t^{10}-175196160z^{2}w^{4}t^{12}-15298560z^{2}w^{2}t^{14}+1351680z^{2}t^{16}-9111771w^{18}-55430244w^{16}t^{2}-124217712w^{14}t^{4}-99395856w^{12}t^{6}+60317568w^{10}t^{8}+178354944w^{8}t^{10}+135370496w^{6}t^{12}+43219968w^{4}t^{14}+4153344w^{2}t^{16}-192512t^{18}}{t^{4}(3w^{2}+4t^{2})(3645xzw^{10}+12150xzw^{8}t^{2}+4050xzw^{6}t^{4}-24300xzw^{4}t^{6}-27000xzw^{2}t^{8}-5280xzt^{10}+3645z^{2}w^{10}+12150z^{2}w^{8}t^{2}+4050z^{2}w^{6}t^{4}-24300z^{2}w^{4}t^{6}-27000z^{2}w^{2}t^{8}-5280z^{2}t^{10}-729w^{12}-3159w^{10}t^{2}-3159w^{8}t^{4}+4374w^{6}t^{6}+10611w^{4}t^{8}+6360w^{2}t^{10}+752t^{12})}$

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
20.72.3.bj.1	$20$	$2$	$2$	$3$	$1$	$1^{2}$
60.72.1.bb.1	$60$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
60.72.1.cg.2	$60$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
60.72.1.dr.2	$60$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
60.72.3.nf.2	$60$	$2$	$2$	$3$	$0$	$1^{2}$
60.72.3.ou.1	$60$	$2$	$2$	$3$	$2$	$2$
60.72.3.qy.1	$60$	$2$	$2$	$3$	$1$	$1^{2}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.432.29.bzb.2	$60$	$3$	$3$	$29$	$4$	$1^{12}\cdot2^{6}$
60.576.33.kk.1	$60$	$4$	$4$	$33$	$8$	$1^{14}\cdot2^{7}$
60.720.37.hz.1	$60$	$5$	$5$	$37$	$7$	$1^{16}\cdot2^{8}$