Modular curve 120.144.3-60.cb.1.8

• Label: 120.144.3-60.cb.1.8
• Level: $120$
• Index: $144$
• Genus: $3$
• Cusps: $8$
• $\Q$-cusps: $4$

Invariants

Level:	$120$		$\SL_2$-level:	$20$		Newform level:	$720$
Index:	$144$		$\PSL_2$-index:	$72$
Genus:	$3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (of which $4$ are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$		Cusp orbits	$1^{4}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 3$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 3$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

20I3

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}13&100\\78&71\end{bmatrix}$, $\begin{bmatrix}29&40\\42&73\end{bmatrix}$, $\begin{bmatrix}61&110\\26&53\end{bmatrix}$, $\begin{bmatrix}71&70\\96&29\end{bmatrix}$, $\begin{bmatrix}77&100\\16&93\end{bmatrix}$, $\begin{bmatrix}89&20\\32&43\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.72.3.cb.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 $\mathbb{P}^{4}$

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

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{4} y - x^{4} z - 3 x^{2} y^{2} z - 6 x^{2} y z^{2} + 6 x^{2} z^{3} + 45 y z^{4} - 45 z^{5} $

Weierstrass model Weierstrass model

$ y^{2} + \left(x^{4} + 1\right) y $

$=$

$ 6x^{8} - 30x^{6} + 49x^{4} - 54x^{2} + 20 $

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model

$(0:0:0:1:0)$, $(1:0:0:0:0)$, $(1:1:1:0:0)$, $(0:0:1:0:0)$

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle -\frac{151875x^{11}-759375x^{9}w^{2}+2784375x^{9}wt-1518750x^{9}t^{2}-13584375x^{7}w^{2}t^{2}+44836875x^{7}wt^{3}-62151057x^{7}t^{4}-10813851x^{5}w^{2}t^{4}+90359775x^{5}wt^{5}-11020266x^{5}t^{6}-110571375x^{3}w^{2}t^{6}+180980643x^{3}wt^{7}+10595772x^{3}t^{8}+28296573xw^{2}t^{8}-66017343xwt^{9}+3096576xt^{10}+1518750yz^{10}-7036875yz^{8}t^{2}+22184307yz^{6}t^{4}+28761120yz^{4}t^{6}-89162721yz^{2}t^{8}-133170044yt^{10}+759375z^{11}+2480625z^{9}t^{2}+1525500z^{7}t^{4}+12449844z^{5}t^{6}+10268217z^{3}t^{8}+16zw^{10}-176zw^{9}t+12400zw^{8}t^{2}-152528zw^{7}t^{3}-25384331zw^{6}t^{4}+47006772zw^{5}t^{5}-136407182zw^{4}t^{6}-100447405zw^{3}t^{7}+49590839zw^{2}t^{8}-8404992zwt^{9}+144695960zt^{10}}{t^{2}(144x^{5}wt^{3}-2880x^{5}t^{4}+7773x^{3}w^{2}t^{4}-15897x^{3}wt^{5}+8121x^{3}t^{6}-7333xw^{2}t^{6}+10530xwt^{7}-1792xt^{8}+3375yz^{6}t^{2}+5661yz^{4}t^{4}+7146yz^{2}t^{6}+9560yt^{8}-3375z^{7}t^{2}-4050z^{5}t^{4}+417z^{3}t^{6}-16zw^{8}+208zw^{7}t-1056zw^{6}t^{2}+2080zw^{5}t^{3}+4144zw^{4}t^{4}+18280zw^{3}t^{5}-18105zw^{2}t^{6}+4864zwt^{7}-8236zt^{8})}$

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

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{3}w$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{3}t$

Equation of the image curve:

$0$

$=$

$ X^{4}Y-X^{4}Z-3X^{2}Y^{2}Z-6X^{2}YZ^{2}+6X^{2}Z^{3}+45YZ^{4}-45Z^{5} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 60.72.3.cb.1 :

$\displaystyle X$	$=$	$\displaystyle -t$
$\displaystyle Y$	$=$	$\displaystyle 4y^{4}-3y^{2}wt-3y^{2}t^{2}+2t^{4}$
$\displaystyle Z$	$=$	$\displaystyle y$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.72.1-10.a.1.1	$40$	$2$	$2$	$1$	$0$
120.72.1-10.a.1.1	$120$	$2$	$2$	$1$	$?$

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.y.1.6	$120$	$2$	$2$	$5$
120.288.5-60.bb.2.3	$120$	$2$	$2$	$5$
120.288.5-60.bc.1.2	$120$	$2$	$2$	$5$
120.288.5-60.bf.2.21	$120$	$2$	$2$	$5$
120.288.5-60.bk.2.7	$120$	$2$	$2$	$5$
120.288.5-60.bn.1.2	$120$	$2$	$2$	$5$
120.288.5-60.bo.2.2	$120$	$2$	$2$	$5$
120.288.5-60.br.1.4	$120$	$2$	$2$	$5$
120.288.5-120.cw.1.8	$120$	$2$	$2$	$5$
120.288.5-120.df.2.1	$120$	$2$	$2$	$5$
120.288.5-120.di.1.6	$120$	$2$	$2$	$5$
120.288.5-120.dr.2.5	$120$	$2$	$2$	$5$
120.288.5-120.eg.2.7	$120$	$2$	$2$	$5$
120.288.5-120.ep.1.5	$120$	$2$	$2$	$5$
120.288.5-120.es.2.3	$120$	$2$	$2$	$5$
120.288.5-120.fb.1.7	$120$	$2$	$2$	$5$
120.288.7-60.ek.2.1	$120$	$2$	$2$	$7$
120.288.7-60.ek.2.5	$120$	$2$	$2$	$7$
120.288.7-60.el.2.4	$120$	$2$	$2$	$7$
120.288.7-60.el.2.29	$120$	$2$	$2$	$7$
120.288.7-60.en.2.1	$120$	$2$	$2$	$7$
120.288.7-60.en.2.9	$120$	$2$	$2$	$7$
120.288.7-60.eo.2.1	$120$	$2$	$2$	$7$
120.288.7-60.eo.2.13	$120$	$2$	$2$	$7$
120.288.7-60.er.1.4	$120$	$2$	$2$	$7$
120.288.7-60.er.1.16	$120$	$2$	$2$	$7$
120.288.7-60.es.2.1	$120$	$2$	$2$	$7$
120.288.7-60.es.2.9	$120$	$2$	$2$	$7$
120.288.7-60.eu.2.1	$120$	$2$	$2$	$7$
120.288.7-60.eu.2.13	$120$	$2$	$2$	$7$
120.288.7-60.ev.2.1	$120$	$2$	$2$	$7$
120.288.7-60.ev.2.9	$120$	$2$	$2$	$7$
120.288.7-120.bcq.1.9	$120$	$2$	$2$	$7$
120.288.7-120.bcq.1.25	$120$	$2$	$2$	$7$
120.288.7-120.bct.2.3	$120$	$2$	$2$	$7$
120.288.7-120.bct.2.27	$120$	$2$	$2$	$7$
120.288.7-120.bcy.1.9	$120$	$2$	$2$	$7$
120.288.7-120.bcy.1.25	$120$	$2$	$2$	$7$
120.288.7-120.bdb.2.3	$120$	$2$	$2$	$7$
120.288.7-120.bdb.2.27	$120$	$2$	$2$	$7$
120.288.7-120.bdj.2.5	$120$	$2$	$2$	$7$
120.288.7-120.bdj.2.29	$120$	$2$	$2$	$7$
120.288.7-120.bdm.1.3	$120$	$2$	$2$	$7$
120.288.7-120.bdm.1.19	$120$	$2$	$2$	$7$
120.288.7-120.bdr.2.5	$120$	$2$	$2$	$7$
120.288.7-120.bdr.2.29	$120$	$2$	$2$	$7$
120.288.7-120.bdu.1.3	$120$	$2$	$2$	$7$
120.288.7-120.bdu.1.19	$120$	$2$	$2$	$7$
120.432.15-60.v.1.4	$120$	$3$	$3$	$15$