Modular curve 60.24.1.d.1

• Label: 60.24.1.d.1
• Level: $60$
• Index: $24$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $4$
• $\Q$-cusps: $0$

Invariants

Level:	$60$		$\SL_2$-level:	$10$		Newform level:	$20$
Index:	$24$		$\PSL_2$-index:	$24$
Genus:	$1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (none of which are rational)		Cusp widths	$2^{2}\cdot10^{2}$		Cusp orbits	$2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ 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:	10D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.24.1.28

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}9&40\\10&23\end{bmatrix}$, $\begin{bmatrix}33&20\\56&9\end{bmatrix}$, $\begin{bmatrix}44&15\\25&29\end{bmatrix}$, $\begin{bmatrix}44&55\\23&41\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	60.48.1-60.d.1.1, 60.48.1-60.d.1.2, 60.48.1-60.d.1.3, 60.48.1-60.d.1.4, 120.48.1-60.d.1.1, 120.48.1-60.d.1.2, 120.48.1-60.d.1.3, 120.48.1-60.d.1.4
Cyclic 60-isogeny field degree:	$24$
Cyclic 60-torsion field degree:	$384$
Full 60-torsion field degree:	$92160$

Jacobian

Conductor:	$2^{2}\cdot5$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	20.2.a.a

Models

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

$ 0 $	$=$	$ 15 x^{2} - y w $
	$=$	$125 y^{2} + 22 y w - 15 z^{2} + w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 25 x^{4} + 66 x^{2} z^{2} - 3 y^{2} z^{2} + 45 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:

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle z$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{15}w$

Maps to other modular curves

$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 3^3\,\frac{89062500yz^{4}w+869820000yz^{2}w^{3}+4376384yw^{5}+1953125z^{6}+171825000z^{4}w^{2}+119227920z^{2}w^{4}+329472w^{6}}{w(3515625yz^{4}+2253750yz^{2}w^{2}+68381yw^{4}-1237500z^{4}w+5280z^{2}w^{3}+5148w^{5})}$

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
10.12.1.a.1	$10$	$2$	$2$	$1$	$0$	dimension zero
60.12.0.bl.1	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.12.0.bo.1	$60$	$2$	$2$	$0$	$0$	full Jacobian

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.72.1.b.1	$60$	$3$	$3$	$1$	$0$	dimension zero
60.72.5.d.2	$60$	$3$	$3$	$5$	$0$	$1^{2}\cdot2$
60.96.5.d.2	$60$	$4$	$4$	$5$	$0$	$1^{2}\cdot2$
60.96.5.l.2	$60$	$4$	$4$	$5$	$0$	$1^{2}\cdot2$
60.120.5.o.1	$60$	$5$	$5$	$5$	$0$	$1^{2}\cdot2$
180.72.1.c.2	$180$	$3$	$3$	$1$	$?$	dimension zero
300.120.5.a.1	$300$	$5$	$5$	$5$	$?$	not computed