Modular curve 60.40.1.z.1

• Label: 60.40.1.z.1
• Level: $60$
• Index: $40$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $4$
• $\Q$-cusps: $0$

Invariants

Level:	$60$		$\SL_2$-level:	$10$		Newform level:	$720$
Index:	$40$		$\PSL_2$-index:	$40$
Genus:	$1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (none of which are rational)		Cusp widths	$10^{4}$		Cusp orbits	$4$
Elliptic points:	$0$ of order $2$ and $4$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	10H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.40.1.23

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}9&34\\44&11\end{bmatrix}$, $\begin{bmatrix}11&53\\18&35\end{bmatrix}$, $\begin{bmatrix}49&7\\55&36\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 60-isogeny field degree:	$144$
Cyclic 60-torsion field degree:	$2304$
Full 60-torsion field degree:	$55296$

Jacobian

Conductor:	$2^{4}\cdot3^{2}\cdot5$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	720.2.a.h

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $	$=$	$ 5 x^{2} + y^{2} + y z + y w - z^{2} - w^{2} $
	$=$	$3 y^{2} + y z + 2 y w - 2 z^{2} + 2 z w - 3 w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{4} - 5 x^{3} z + 15 x^{2} y^{2} + 10 x z^{3} - 100 y^{4} - 5 z^{4} $

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 embedded model to plane model:

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle x$
$\displaystyle Z$	$=$	$\displaystyle w$

Maps to other modular curves

$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 3\cdot5^2\,\frac{8470288yz^{9}-10149600yz^{8}w+21969096yz^{7}w^{2}-10982400yz^{6}w^{3}+13925229yz^{5}w^{4}-972390yz^{4}w^{5}+3355722yz^{3}w^{6}+877740yz^{2}w^{7}+489840yzw^{8}+80600yw^{9}-5701472z^{10}+12525216z^{9}w-26517984z^{8}w^{2}+25888752z^{7}w^{3}-26180646z^{6}w^{4}+11840178z^{5}w^{5}-7963203z^{4}w^{6}+706224z^{3}w^{7}-1013595z^{2}w^{8}-252520zw^{9}-61500w^{10}}{11810yz^{9}+63765yz^{8}w+158850yz^{7}w^{2}+246990yz^{6}w^{3}+277065yz^{5}w^{4}+204345yz^{4}w^{5}+92130yz^{3}w^{6}+24075yz^{2}w^{7}+3330yzw^{8}+205yw^{9}-7873z^{10}-34650z^{9}w-70260z^{8}w^{2}-101640z^{7}w^{3}-121560z^{6}w^{4}-142062z^{5}w^{5}-116325z^{4}w^{6}-56445z^{3}w^{7}-15855z^{2}w^{8}-2420zw^{9}-132w^{10}}$

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
15.20.0.a.1	$15$	$2$	$2$	$0$	$0$	full Jacobian
20.20.0.c.1	$20$	$2$	$2$	$0$	$0$	full Jacobian
60.20.1.b.1	$60$	$2$	$2$	$1$	$0$	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
60.120.5.dk.1	$60$	$3$	$3$	$5$	$0$	$1^{4}$
60.120.5.er.1	$60$	$3$	$3$	$5$	$1$	$1^{4}$
60.120.7.em.1	$60$	$3$	$3$	$7$	$2$	$1^{4}\cdot2$
60.120.9.ee.1	$60$	$3$	$3$	$9$	$2$	$1^{6}\cdot2$
60.160.9.cg.1	$60$	$4$	$4$	$9$	$3$	$1^{8}$
60.160.9.de.1	$60$	$4$	$4$	$9$	$5$	$1^{8}$
300.200.9.bm.1	$300$	$5$	$5$	$9$	$?$	not computed