Modular curve 60.288.5-60.pa.1.7

• Label: 60.288.5-60.pa.1.7
• Level: $60$
• Index: $288$
• Genus: $5$
• Analytic rank: $2$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$60$		$\SL_2$-level:	$20$		Newform level:	$3600$
Index:	$288$		$\PSL_2$-index:	$144$
Genus:	$5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)		Cusp widths	$2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$		Cusp orbits	$2^{4}\cdot4^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$2$
$\Q$-gonality:	$4$
$\overline{\Q}$-gonality:	$4$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	20I5
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.288.5.980

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}7&20\\16&21\end{bmatrix}$, $\begin{bmatrix}29&15\\6&29\end{bmatrix}$, $\begin{bmatrix}33&20\\38&1\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.144.5.pa.1 for the level structure with $-I$)
Cyclic 60-isogeny field degree:	$8$
Cyclic 60-torsion field degree:	$64$
Full 60-torsion field degree:	$7680$

Jacobian

Conductor:	$2^{17}\cdot3^{8}\cdot5^{7}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{3}\cdot2$
Newforms:	360.2.a.a, 360.2.f.c, 400.2.a.c, 3600.2.a.be

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $	$=$	$ x^{2} - y z $
	$=$	$2 x^{2} + 5 x y - 5 x z + 3 y z + t^{2}$
	$=$	$5 x^{2} + 5 y^{2} + 5 y z + 5 z^{2} - 3 w^{2} + 4 t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 10 x^{8} - 30 x^{7} y - 21 x^{6} y^{2} + 900 x^{6} z^{2} + 18 x^{5} y^{3} - 1050 x^{5} y z^{2} + \cdots + 255625 z^{8} $

Rational points

This modular curve has no $\Q_p$ points for $p=19$, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{1}{3}\cdot\frac{512545320yzw^{16}-3075271920yzw^{14}t^{2}+6717414240yzw^{12}t^{4}-6749256960yzw^{10}t^{6}+3144614400yzw^{8}t^{8}-446653440yzw^{6}t^{10}-297768960yzw^{4}t^{12}+269291520yzw^{2}t^{14}-79994880yzt^{16}-61509375w^{18}+492075000w^{16}t^{2}-1510394688w^{14}t^{4}+2287228752w^{12}t^{6}-1815478272w^{10}t^{8}+732810240w^{8}t^{10}-129358080w^{6}t^{12}-497664w^{4}t^{14}+8257536w^{2}t^{16}-3198976t^{18}}{t^{4}w^{2}(1215yzw^{10}-4050yzw^{8}t^{2}+1350yzw^{6}t^{4}+900yzw^{4}t^{6}+600yzw^{2}t^{8}+320yzt^{10}-81w^{8}t^{4}+108w^{6}t^{6}+81w^{4}t^{8}+72w^{2}t^{10}+64t^{12})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.5.pa.1 :

$\displaystyle X$	$=$	$\displaystyle x+y-z$
$\displaystyle Y$	$=$	$\displaystyle z+w$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{5}t$

Equation of the image curve:

$0$

$=$

$ 10X^{8}-30X^{7}Y-21X^{6}Y^{2}+18X^{5}Y^{3}+9X^{4}Y^{4}+900X^{6}Z^{2}-1050X^{5}YZ^{2}-810X^{4}Y^{2}Z^{2}+90X^{3}Y^{3}Z^{2}+21250X^{4}Z^{4}-8700X^{3}YZ^{4}-3975X^{2}Y^{2}Z^{4}+135000X^{2}Z^{6}-21000XYZ^{6}+255625Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
20.144.1-20.k.2.7	$20$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
60.144.1-20.k.2.6	$60$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
60.144.1-60.cf.1.2	$60$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
60.144.1-60.cf.1.3	$60$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
60.144.3-60.ys.2.13	$60$	$2$	$2$	$3$	$1$	$1^{2}$
60.144.3-60.ys.2.14	$60$	$2$	$2$	$3$	$1$	$1^{2}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.864.29-60.dwi.2.9	$60$	$3$	$3$	$29$	$2$	$1^{12}\cdot2^{6}$
60.1152.33-60.nn.2.7	$60$	$4$	$4$	$33$	$7$	$1^{14}\cdot2^{7}$
60.1440.37-60.nv.1.3	$60$	$5$	$5$	$37$	$9$	$1^{16}\cdot2^{8}$