Modular curve 120.144.3-60.fc.1.14

• Label: 120.144.3-60.fc.1.14
• Level: $120$
• Index: $144$
• Genus: $3$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$120$		$\SL_2$-level:	$40$		Newform level:	$3600$
Index:	$144$		$\PSL_2$-index:	$72$
Genus:	$3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$		Cusp orbits	$2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 4$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 3$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

20J3

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}7&50\\0&97\end{bmatrix}$, $\begin{bmatrix}63&50\\23&3\end{bmatrix}$, $\begin{bmatrix}89&0\\80&101\end{bmatrix}$, $\begin{bmatrix}89&20\\24&13\end{bmatrix}$, $\begin{bmatrix}111&70\\119&87\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.72.3.fc.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree:	$16$
Cyclic 120-torsion field degree:	$512$
Full 120-torsion field degree:	$245760$

Models

Embedded model Embedded model in $\mathbb{P}^{5}$

$ 0 $	$=$	$ - x w + 2 y z $
	$=$	$x t + y z + 2 y t + 2 y u$
	$=$	$z w + 2 z t + 2 w t + 2 w u$
	$=$	$x^{2} + 4 y^{2} + z^{2} + 2 z w$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 900 x^{6} + 3255 x^{4} y^{2} + 10335 x^{4} y z + 7980 x^{4} z^{2} + 4 x^{2} y^{4} - 239 x^{2} y^{3} z + \cdots + 4 z^{6} $

Geometric Weierstrass model Geometric Weierstrass model

$ 25 w^{2} $	$=$	$ 225 x^{4} + 45 x^{2} z^{2} + z^{4} $
$0$	$=$	$15 x^{2} + y^{2} + z^{2}$

Rational points

This modular curve has no real points, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{1}{2^4}\cdot\frac{1166400000z^{10}-10874250000z^{9}u+61357500000z^{8}u^{2}-15430500000z^{7}u^{3}+33728400000z^{6}u^{4}-6282360000z^{5}u^{5}-6990192000z^{4}u^{6}+3964838400z^{3}u^{7}-145263360z^{2}u^{8}+2006100373504zt^{9}+7741319070464zt^{8}u+16609634777920zt^{7}u^{2}+13177863972224zt^{6}u^{3}-6203267733160zt^{5}u^{4}-10522750831751zt^{4}u^{5}+2003461932403zt^{3}u^{6}+5521606486856zt^{2}u^{7}+1520726460994ztu^{8}-1275196672zu^{9}+2071260135424wt^{9}+9958687590400wt^{8}u+24845995776000wt^{7}u^{2}+30176536661504wt^{6}u^{3}+6698478811056wt^{5}u^{4}-17134783346192wt^{4}u^{5}-8389094541315wt^{3}u^{6}+7772350983191wt^{2}u^{7}+7061841296888wtu^{8}+1498148026650wu^{9}+172871569408t^{10}+915169112576t^{9}u+2477687534208t^{8}u^{2}+3541342076416t^{7}u^{3}+1939590904640t^{6}u^{4}-845901066562t^{5}u^{5}-1356073838216t^{4}u^{6}+75017986825t^{3}u^{7}+854869091881t^{2}u^{8}+517171039524tu^{9}+101737299358u^{10}}{(t+u)^{2}(14613158400zt^{7}+54848217600zt^{6}u+84603467700zt^{5}u^{2}+68387076300zt^{4}u^{3}+30388966200zt^{3}u^{4}+6982689825zt^{2}u^{5}+639880800ztu^{6}+15035718400wt^{7}+71109940480wt^{6}u+141953686560wt^{5}u^{2}+154755230000wt^{4}u^{3}+99224272400wt^{3}u^{4}+37261152000wt^{2}u^{5}+7540910080wtu^{6}+628264960wu^{7}+1283245056t^{8}+6527317888t^{7}u+14639997928t^{6}u^{2}+18992393496t^{5}u^{3}+15615102620t^{4}u^{4}+8314527086t^{3}u^{5}+2780846658t^{2}u^{6}+527612608tu^{7}+42658816u^{8})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.72.3.fc.1 :

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle u$
$\displaystyle Z$	$=$	$\displaystyle t$

Equation of the image curve:

$0$

$=$

$ 900X^{6}+3255X^{4}Y^{2}+4X^{2}Y^{4}+10335X^{4}YZ-239X^{2}Y^{3}Z+7980X^{4}Z^{2}-411X^{2}Y^{2}Z^{2}+4Y^{4}Z^{2}+136X^{2}YZ^{3}+Y^{3}Z^{3}+304X^{2}Z^{4}-6Y^{2}Z^{4}+YZ^{5}+4Z^{6} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.72.1-20.b.1.9	$40$	$2$	$2$	$1$	$0$
120.24.0-60.e.1.6	$120$	$6$	$6$	$0$	$?$
120.72.1-20.b.1.11	$120$	$2$	$2$	$1$	$?$

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

Covering curve	Level	Index	Degree	Genus
120.288.5-60.fo.1.12	$120$	$2$	$2$	$5$
120.288.5-60.fo.2.10	$120$	$2$	$2$	$5$
120.288.5-60.fp.1.8	$120$	$2$	$2$	$5$
120.288.5-60.fp.2.6	$120$	$2$	$2$	$5$
120.288.5-60.fw.1.6	$120$	$2$	$2$	$5$
120.288.5-60.fw.2.2	$120$	$2$	$2$	$5$
120.288.5-60.fx.1.4	$120$	$2$	$2$	$5$
120.288.5-60.fx.2.2	$120$	$2$	$2$	$5$
120.288.5-120.boa.1.3	$120$	$2$	$2$	$5$
120.288.5-120.boa.2.3	$120$	$2$	$2$	$5$
120.288.5-120.boh.1.3	$120$	$2$	$2$	$5$
120.288.5-120.boh.2.3	$120$	$2$	$2$	$5$
120.288.5-120.bqe.1.5	$120$	$2$	$2$	$5$
120.288.5-120.bqe.2.3	$120$	$2$	$2$	$5$
120.288.5-120.bql.1.5	$120$	$2$	$2$	$5$
120.288.5-120.bql.2.3	$120$	$2$	$2$	$5$
120.432.15-60.ce.1.5	$120$	$3$	$3$	$15$