Modular curve 120.120.4-60.c.1.6

• Label: 120.120.4-60.c.1.6
• Level: $120$
• Index: $120$
• Genus: $4$
• Cusps: $4$
• $\Q$-cusps: $2$

Invariants

Level:	$120$		$\SL_2$-level:	$20$		Newform level:	$3600$
Index:	$120$		$\PSL_2$-index:	$60$
Genus:	$4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (of which $2$ are rational)		Cusp widths	$10^{2}\cdot20^{2}$		Cusp orbits	$1^{2}\cdot2$
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:	$2 \le \gamma \le 4$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

20A4

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}25&24\\102&85\end{bmatrix}$, $\begin{bmatrix}27&62\\34&97\end{bmatrix}$, $\begin{bmatrix}59&82\\66&61\end{bmatrix}$, $\begin{bmatrix}79&70\\116&67\end{bmatrix}$, $\begin{bmatrix}91&100\\92&67\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.60.4.c.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree:	$96$
Cyclic 120-torsion field degree:	$3072$
Full 120-torsion field degree:	$294912$

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $	$=$	$ 30 x^{2} + 15 x y + 15 y^{2} - z w - w^{2} $
	$=$	$15 x y^{2} + x z^{2} + x z w - 15 y^{3} + y z^{2} + 3 y z w + y w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ - 13500 x^{6} - 1125 x^{4} y^{2} - 225 x^{4} y z + 1350 x^{4} z^{2} - 30 x^{2} y^{4} - 45 x^{2} y^{3} z + \cdots + y^{2} z^{4} $

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

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

Maps to other modular curves

$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 2^4\,\frac{84675xyz^{8}+245055xyz^{7}w-368640xyz^{6}w^{2}-2421150xyz^{5}w^{3}-2893725xyz^{4}w^{4}+1143435xyz^{3}w^{5}+4325670xyz^{2}w^{6}+2685540xyzw^{7}+488280xyw^{8}+17145y^{2}z^{8}+115365y^{2}z^{7}w+49560y^{2}z^{6}w^{2}-1290330y^{2}z^{5}w^{3}-3324675y^{2}z^{4}w^{4}-2446905y^{2}z^{3}w^{5}+907710y^{2}z^{2}w^{6}+1708980y^{2}zw^{7}+488280y^{2}w^{8}+2048z^{10}+12491z^{9}w+29060z^{8}w^{2}+25933z^{7}w^{3}-2390z^{6}w^{4}+7291z^{5}w^{5}+89468z^{4}w^{6}+131925z^{3}w^{7}+78642z^{2}w^{8}+17356zw^{9}+216w^{10}}{255xyz^{8}+1125xyz^{7}w+1725xyz^{6}w^{2}+1665xyz^{5}w^{3}+975xyz^{4}w^{4}+15xyz^{3}w^{5}-345xyz^{2}w^{6}-165xyzw^{7}-30xyw^{8}+45y^{2}z^{8}+375y^{2}z^{7}w+495y^{2}z^{6}w^{2}+75y^{2}z^{5}w^{3}-75y^{2}z^{4}w^{4}-195y^{2}z^{3}w^{5}-135y^{2}z^{2}w^{6}-105y^{2}zw^{7}-30y^{2}w^{8}+7z^{9}w+30z^{8}w^{2}+60z^{7}w^{3}+68z^{6}w^{4}+34z^{5}w^{5}-8z^{4}w^{6}-10z^{3}w^{7}+8z^{2}w^{8}+9zw^{9}+2w^{10}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.60.4.c.1 :

$\displaystyle X$	$=$	$\displaystyle x+y$
$\displaystyle Y$	$=$	$\displaystyle z$
$\displaystyle Z$	$=$	$\displaystyle w$

Equation of the image curve:

$0$

$=$

$ -13500X^{6}-1125X^{4}Y^{2}-225X^{4}YZ+1350X^{4}Z^{2}-30X^{2}Y^{4}-45X^{2}Y^{3}Z+15X^{2}YZ^{3}-30X^{2}Z^{4}+Y^{3}Z^{3}+Y^{2}Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.60.2-10.a.1.2	$40$	$2$	$2$	$2$	$0$
120.24.0-60.a.1.5	$120$	$5$	$5$	$0$	$?$
120.60.2-10.a.1.3	$120$	$2$	$2$	$2$	$?$

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

Covering curve	Level	Index	Degree	Genus
120.240.8-60.d.1.11	$120$	$2$	$2$	$8$
120.240.8-60.f.1.14	$120$	$2$	$2$	$8$
120.240.8-60.f.1.15	$120$	$2$	$2$	$8$
120.240.8-60.g.1.2	$120$	$2$	$2$	$8$
120.240.8-60.g.1.8	$120$	$2$	$2$	$8$
120.240.8-120.i.1.4	$120$	$2$	$2$	$8$
120.240.8-120.i.1.15	$120$	$2$	$2$	$8$
120.240.8-60.j.1.2	$120$	$2$	$2$	$8$
120.240.8-60.j.1.8	$120$	$2$	$2$	$8$
120.240.8-120.m.1.4	$120$	$2$	$2$	$8$
120.240.8-120.m.1.14	$120$	$2$	$2$	$8$
120.240.8-120.q.1.2	$120$	$2$	$2$	$8$
120.240.8-120.q.1.16	$120$	$2$	$2$	$8$
120.240.8-120.z.1.2	$120$	$2$	$2$	$8$
120.240.8-120.z.1.16	$120$	$2$	$2$	$8$
120.360.10-60.c.1.8	$120$	$3$	$3$	$10$
120.360.14-60.g.1.6	$120$	$3$	$3$	$14$
120.480.13-60.bk.1.3	$120$	$4$	$4$	$13$
120.480.17-60.g.1.12	$120$	$4$	$4$	$17$