Modular curve 60.72.1.cf.1

• Label: 60.72.1.cf.1
• Level: $60$
• Index: $72$
• Genus: $1$
• Analytic rank: $1$
• Cusps: $12$
• $\Q$-cusps: $2$

Invariants

Level:	$60$		$\SL_2$-level:	$10$		Newform level:	$3600$
Index:	$72$		$\PSL_2$-index:	$72$
Genus:	$1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $2$ are rational)		Cusp widths	$2^{6}\cdot10^{6}$		Cusp orbits	$1^{2}\cdot2^{3}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	10K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.72.1.26

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}13&45\\28&59\end{bmatrix}$, $\begin{bmatrix}17&10\\52&21\end{bmatrix}$, $\begin{bmatrix}41&50\\14&49\end{bmatrix}$, $\begin{bmatrix}43&40\\38&31\end{bmatrix}$, $\begin{bmatrix}57&5\\58&51\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	60.144.1-60.cf.1.1, 60.144.1-60.cf.1.2, 60.144.1-60.cf.1.3, 60.144.1-60.cf.1.4, 60.144.1-60.cf.1.5, 60.144.1-60.cf.1.6, 60.144.1-60.cf.1.7, 60.144.1-60.cf.1.8, 60.144.1-60.cf.1.9, 60.144.1-60.cf.1.10, 60.144.1-60.cf.1.11, 60.144.1-60.cf.1.12, 60.144.1-60.cf.1.13, 60.144.1-60.cf.1.14, 60.144.1-60.cf.1.15, 60.144.1-60.cf.1.16, 120.144.1-60.cf.1.1, 120.144.1-60.cf.1.2, 120.144.1-60.cf.1.3, 120.144.1-60.cf.1.4, 120.144.1-60.cf.1.5, 120.144.1-60.cf.1.6, 120.144.1-60.cf.1.7, 120.144.1-60.cf.1.8, 120.144.1-60.cf.1.9, 120.144.1-60.cf.1.10, 120.144.1-60.cf.1.11, 120.144.1-60.cf.1.12, 120.144.1-60.cf.1.13, 120.144.1-60.cf.1.14, 120.144.1-60.cf.1.15, 120.144.1-60.cf.1.16
Cyclic 60-isogeny field degree:	$8$
Cyclic 60-torsion field degree:	$64$
Full 60-torsion field degree:	$30720$

Jacobian

Conductor:	$2^{4}\cdot3^{2}\cdot5^{2}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	3600.2.a.be

Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

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
$X_{\pm1}(10)$	$10$	$2$	$2$	$0$	$0$	full Jacobian
60.24.1.bd.1	$60$	$3$	$3$	$1$	$1$	dimension zero
60.36.0.e.1	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.36.1.bf.1	$60$	$2$	$2$	$1$	$1$	dimension zero

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	$60$	$2$	$2$	$5$	$1$	$1^{2}\cdot2$
60.144.5.fu.1	$60$	$2$	$2$	$5$	$1$	$1^{2}\cdot2$
60.144.5.kp.1	$60$	$2$	$2$	$5$	$1$	$1^{2}\cdot2$
60.144.5.kv.1	$60$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
60.144.5.pa.1	$60$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
60.144.5.pg.1	$60$	$2$	$2$	$5$	$1$	$1^{2}\cdot2$
60.144.5.qa.1	$60$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
60.144.5.qe.1	$60$	$2$	$2$	$5$	$2$	$1^{2}\cdot2$
60.216.13.gn.2	$60$	$3$	$3$	$13$	$1$	$1^{6}\cdot2^{3}$
60.288.13.nr.2	$60$	$4$	$4$	$13$	$4$	$1^{6}\cdot2^{3}$
60.360.13.bu.1	$60$	$5$	$5$	$13$	$3$	$1^{6}\cdot2^{3}$
120.144.5.bor.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.bpt.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.ddf.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.dev.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.egs.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.eii.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.enq.1	$120$	$2$	$2$	$5$	$?$	not computed
120.144.5.eos.1	$120$	$2$	$2$	$5$	$?$	not computed
300.360.13.bd.1	$300$	$5$	$5$	$13$	$?$	not computed