Modular curve 60.72.3.fd.1

• Label: 60.72.3.fd.1
• Level: $60$
• Index: $72$
• Genus: $3$
• Analytic rank: $1$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$60$		$\SL_2$-level:	$20$		Newform level:	$3600$
Index:	$72$		$\PSL_2$-index:	$72$
Genus:	$3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$		Cusp orbits	$2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$4$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	20J3
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.72.3.745

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}5&44\\14&1\end{bmatrix}$, $\begin{bmatrix}21&56\\14&21\end{bmatrix}$, $\begin{bmatrix}47&30\\8&13\end{bmatrix}$, $\begin{bmatrix}53&2\\19&5\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	120.144.3-60.fd.1.1, 120.144.3-60.fd.1.2, 120.144.3-60.fd.1.3, 120.144.3-60.fd.1.4, 120.144.3-60.fd.1.5, 120.144.3-60.fd.1.6, 120.144.3-60.fd.1.7, 120.144.3-60.fd.1.8, 120.144.3-60.fd.1.9, 120.144.3-60.fd.1.10, 120.144.3-60.fd.1.11, 120.144.3-60.fd.1.12, 120.144.3-60.fd.1.13, 120.144.3-60.fd.1.14, 120.144.3-60.fd.1.15, 120.144.3-60.fd.1.16
Cyclic 60-isogeny field degree:	$8$
Cyclic 60-torsion field degree:	$128$
Full 60-torsion field degree:	$30720$

Jacobian

Conductor:	$2^{11}\cdot3^{4}\cdot5^{5}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{3}$
Newforms:	80.2.a.b, 1800.2.a.v, 3600.2.a.be

Models

Embedded model Embedded model in $\mathbb{P}^{5}$

$ 0 $	$=$	$ - x w + 2 y z $
	$=$	$z u + 2 w t - 2 w u$
	$=$	$x^{2} + 4 y^{2} + z^{2} - 2 z w$
	$=$	$x u + 4 y t - 4 y u$
	$=$	$\cdots$

Geometric Weierstrass model Geometric Weierstrass model

$ 25 w^{2} $	$=$	$ -225 x^{4} + 15 x^{2} z^{2} + z^{4} $
$0$	$=$	$-15 x^{2} + y^{2} + z^{2}$

Rational points

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

Maps to other modular curves

$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 2^6\cdot3\,\frac{10800000z^{8}t^{2}+10800000z^{8}tu+5760000z^{6}t^{2}u^{2}-3240000z^{6}tu^{3}+630000z^{6}u^{4}-474000z^{4}t^{2}u^{4}+783000z^{4}tu^{5}+104625z^{4}u^{6}-66700z^{2}t^{2}u^{6}+110660z^{2}tu^{7}-79985z^{2}u^{8}-118125w^{8}u^{2}+1125w^{6}u^{4}+21900w^{4}u^{6}+44085w^{2}u^{8}+364t^{2}u^{8}+424tu^{9}-409u^{10}}{u^{4}(540000z^{6}-288000z^{4}t^{2}+468000z^{4}tu-63000z^{4}u^{2}-19800z^{2}t^{2}u^{2}+44100z^{2}tu^{3}-20955z^{2}u^{4}-6750w^{6}-1575w^{4}u^{2}-180w^{2}u^{4}-44t^{2}u^{4}+216tu^{5}-151u^{6})}$

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
20.36.1.b.1	$20$	$2$	$2$	$1$	$0$	$1^{2}$
60.12.0.f.1	$60$	$6$	$6$	$0$	$0$	full Jacobian
60.36.1.bf.1	$60$	$2$	$2$	$1$	$1$	$1^{2}$
60.36.1.dm.1	$60$	$2$	$2$	$1$	$0$	$1^{2}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.144.5.fq.1	$60$	$2$	$2$	$5$	$1$	$2$
60.144.5.fq.2	$60$	$2$	$2$	$5$	$1$	$2$
60.144.5.fr.1	$60$	$2$	$2$	$5$	$1$	$2$
60.144.5.fr.2	$60$	$2$	$2$	$5$	$1$	$2$
60.144.5.fy.1	$60$	$2$	$2$	$5$	$1$	$2$
60.144.5.fy.2	$60$	$2$	$2$	$5$	$1$	$2$
60.144.5.fz.1	$60$	$2$	$2$	$5$	$1$	$2$
60.144.5.fz.2	$60$	$2$	$2$	$5$	$1$	$2$
60.216.15.cf.1	$60$	$3$	$3$	$15$	$3$	$1^{12}$
60.288.17.bh.1	$60$	$4$	$4$	$17$	$7$	$1^{14}$
60.360.19.fq.1	$60$	$5$	$5$	$19$	$6$	$1^{16}$
120.144.5.boo.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.boo.2	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.bov.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.bov.2	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.bqs.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.bqs.2	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.bqz.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.bqz.2	$120$	$2$	$2$	$5$	$?$	not computed
300.360.19.r.1	$300$	$5$	$5$	$19$	$?$	not computed