Modular curve 120.288.5-60.pg.1.5

• Label: 120.288.5-60.pg.1.5
• 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	$2^{6}\cdot4$
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:	$\begin{bmatrix}23&110\\69&31\end{bmatrix}$, $\begin{bmatrix}49&20\\97&49\end{bmatrix}$, $\begin{bmatrix}53&70\\10&9\end{bmatrix}$, $\begin{bmatrix}61&80\\2&89\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.144.5.pg.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} $
	$=$	$2 x y + 5 x z - 2 y^{2} - 3 z^{2} + t^{2}$
	$=$	$5 x^{2} - 3 x y + 10 x z + 3 y^{2} + 7 z^{2} - 3 w^{2} - 2 t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 7290 x^{8} + 2430 x^{7} y - 19440 x^{7} z - 189 x^{6} y^{2} - 9720 x^{6} y z - 50220 x^{6} z^{2} + \cdots + 15665 z^{8} $

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 $X(1)$ :

$\displaystyle j$

$=$

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

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

$\displaystyle X$	$=$	$\displaystyle x-\frac{4}{5}t$
$\displaystyle Y$	$=$	$\displaystyle 9y+9w$
$\displaystyle Z$	$=$	$\displaystyle 3z+\frac{3}{5}t$

Equation of the image curve:

$0$

$=$

$ 7290X^{8}+2430X^{7}Y-189X^{6}Y^{2}-18X^{5}Y^{3}+X^{4}Y^{4}-19440X^{7}Z-9720X^{6}YZ+1008X^{5}Y^{2}Z+120X^{4}Y^{3}Z-8X^{3}Y^{4}Z-50220X^{6}Z^{2}+4320X^{5}YZ^{2}-1080X^{4}Y^{2}Z^{2}-280X^{3}Y^{3}Z^{2}+24X^{2}Y^{4}Z^{2}+130680X^{5}Z^{3}+21600X^{4}YZ^{3}-2240X^{3}Y^{2}Z^{3}+240X^{2}Y^{3}Z^{3}-32XY^{4}Z^{3}+76050X^{4}Z^{4}-18750X^{3}YZ^{4}+4430X^{2}Y^{2}Z^{4}+16Y^{4}Z^{4}-202680X^{3}Z^{5}-10160X^{2}YZ^{5}-632XY^{2}Z^{5}-64Y^{3}Z^{5}-20720X^{2}Z^{6}+7440XYZ^{6}-1416Y^{2}Z^{6}+63440XZ^{7}+2960YZ^{7}+15665Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.144.1-20.m.2.3	$40$	$2$	$2$	$1$	$0$
120.144.1-20.m.2.7	$120$	$2$	$2$	$1$	$?$
120.144.1-60.cf.1.3	$120$	$2$	$2$	$1$	$?$
120.144.1-60.cf.1.6	$120$	$2$	$2$	$1$	$?$
120.144.3-60.yu.2.11	$120$	$2$	$2$	$3$	$?$
120.144.3-60.yu.2.16	$120$	$2$	$2$	$3$	$?$