Modular curve 60.24.1.d.2

• Label: 60.24.1.d.2
• 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:	60.24.1.29

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}1&40\\56&27\end{bmatrix}$, $\begin{bmatrix}17&35\\21&44\end{bmatrix}$, $\begin{bmatrix}41&20\\0&31\end{bmatrix}$, $\begin{bmatrix}46&5\\41&11\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	60.48.1-60.d.2.1, 60.48.1-60.d.2.2, 60.48.1-60.d.2.3, 60.48.1-60.d.2.4, 120.48.1-60.d.2.1, 120.48.1-60.d.2.2, 120.48.1-60.d.2.3, 120.48.1-60.d.2.4
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 $	$=$	$ 6 x^{2} + 5 y^{2} - 4 y w - 3 z^{2} + w^{2} $
	$=$	$9 x^{2} - 5 y^{2} + 5 y w + 3 z^{2} - w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 125 x^{4} + 66 x^{2} z^{2} - 3 y^{2} z^{2} + 9 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:

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle z$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{3}w$

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle 3^3\,\frac{3562500yz^{4}w+173964000yz^{2}w^{3}+4376384yw^{5}-78125z^{6}-34365000z^{4}w^{2}-119227920z^{2}w^{4}-1647360w^{6}}{w(140625yz^{4}+450750yz^{2}w^{2}+68381yw^{4}+247500z^{4}w-5280z^{2}w^{3}-25740w^{5})}$

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.2	$60$	$2$	$2$	$0$	$0$	full Jacobian
60.12.0.bo.2	$60$	$2$	$2$	$0$	$0$	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.2	$60$	$3$	$3$	$1$	$0$	dimension zero
60.72.5.d.1	$60$	$3$	$3$	$5$	$0$	$1^{2}\cdot2$
60.96.5.d.1	$60$	$4$	$4$	$5$	$0$	$1^{2}\cdot2$
60.96.5.l.1	$60$	$4$	$4$	$5$	$0$	$1^{2}\cdot2$
60.120.5.o.1	$60$	$5$	$5$	$5$	$0$	$1^{2}\cdot2$
180.72.1.c.1	$180$	$3$	$3$	$1$	$?$	dimension zero
300.120.5.a.2	$300$	$5$	$5$	$5$	$?$	not computed