Modular curve 120.288.5-60.kp.1.6

• Label: 120.288.5-60.kp.1.6
• Level: $120$
• Index: $288$
• Genus: $5$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$120$		$\SL_2$-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:	not computed
$\Q$-gonality:	$2 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 5$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

20I5

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}13&70\\0&89\end{bmatrix}$, $\begin{bmatrix}47&100\\21&71\end{bmatrix}$, $\begin{bmatrix}73&70\\56&1\end{bmatrix}$, $\begin{bmatrix}73&100\\101&91\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.144.5.kp.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree:	$16$
Cyclic 120-torsion field degree:	$256$
Full 120-torsion field degree:	$122880$

Models

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

$ 0 $	$=$	$ x y + y^{2} - z^{2} $
	$=$	$x y - 3 x z + y^{2} + 2 z^{2} - t^{2}$
	$=$	$3 x^{2} + 2 x y + 6 x z + 2 y^{2} + 4 z^{2} - 5 w^{2} - 2 t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 6 x^{8} + 30 x^{7} y - 48 x^{7} z + 5 x^{6} y^{2} - 300 x^{6} y z + 132 x^{6} z^{2} - 50 x^{5} y^{3} + \cdots + 159 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 $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{1}{5}\cdot\frac{18309375000z^{2}w^{16}+65913750000z^{2}w^{14}t^{2}+86386500000z^{2}w^{12}t^{4}+52077600000z^{2}w^{10}t^{6}+14558400000z^{2}w^{8}t^{8}+1240704000z^{2}w^{6}t^{10}-496281600z^{2}w^{4}t^{12}-269291520z^{2}w^{2}t^{14}-47996928z^{2}t^{16}-6103515625w^{18}-29296875000w^{16}t^{2}-53955000000w^{14}t^{4}-49023250000w^{12}t^{6}-23347200000w^{10}t^{8}-5654400000w^{8}t^{10}-598880000w^{6}t^{12}+1382400w^{4}t^{14}+13762560w^{2}t^{16}+3198976t^{18}}{t^{4}w^{2}(9375z^{2}w^{10}+18750z^{2}w^{8}t^{2}+3750z^{2}w^{6}t^{4}-1500z^{2}w^{4}t^{6}+600z^{2}w^{2}t^{8}-192z^{2}t^{10}-625w^{8}t^{4}-500w^{6}t^{6}+225w^{4}t^{8}-120w^{2}t^{10}+64t^{12})}$

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

$\displaystyle X$	$=$	$\displaystyle x-\frac{2}{3}t$
$\displaystyle Y$	$=$	$\displaystyle y+w$
$\displaystyle Z$	$=$	$\displaystyle z+\frac{1}{3}t$

Equation of the image curve:

$0$

$=$

$ 6X^{8}+30X^{7}Y+5X^{6}Y^{2}-50X^{5}Y^{3}-25X^{4}Y^{4}-48X^{7}Z-300X^{6}YZ-60X^{5}Y^{2}Z+700X^{4}Y^{3}Z+400X^{3}Y^{4}Z+132X^{6}Z^{2}+540X^{5}YZ^{2}-120X^{4}Y^{2}Z^{2}-3500X^{3}Y^{3}Z^{2}-2400X^{2}Y^{4}Z^{2}-120X^{5}Z^{3}+2640X^{4}YZ^{3}+3320X^{3}Y^{2}Z^{3}+6800X^{2}Y^{3}Z^{3}+6400XY^{4}Z^{3}-1146X^{4}Z^{4}-7590X^{3}YZ^{4}-10650X^{2}Y^{2}Z^{4}-1600XY^{3}Z^{4}-6400Y^{4}Z^{4}+4488X^{3}Z^{5}-4860X^{2}YZ^{5}+2640XY^{2}Z^{5}-6400Y^{3}Z^{5}-3288X^{2}Z^{6}+15420XYZ^{6}+19040Y^{2}Z^{6}-2208XZ^{7}+10320YZ^{7}+159Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.144.3-20.bg.2.1	$40$	$2$	$2$	$3$	$0$
120.144.1-60.y.2.7	$120$	$2$	$2$	$1$	$?$
120.144.1-60.y.2.8	$120$	$2$	$2$	$1$	$?$
120.144.1-60.cf.1.1	$120$	$2$	$2$	$1$	$?$
120.144.1-60.cf.1.10	$120$	$2$	$2$	$1$	$?$
120.144.3-20.bg.2.7	$120$	$2$	$2$	$3$	$?$