Modular curve 120.120.4-40.a.1.2

• Label: 120.120.4-40.a.1.2
• Level: $120$
• Index: $120$
• Genus: $4$
• Cusps: $4$
• $\Q$-cusps: $2$

Invariants

Level:	$120$		$\SL_2$-level:	$20$		Newform level:	$1600$
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:	$3 \le \gamma \le 4$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 4$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

20A4

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}7&100\\72&73\end{bmatrix}$, $\begin{bmatrix}17&20\\78&61\end{bmatrix}$, $\begin{bmatrix}19&92\\56&51\end{bmatrix}$, $\begin{bmatrix}31&22\\72&25\end{bmatrix}$, $\begin{bmatrix}119&22\\100&111\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.60.4.a.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 $	$=$	$ 2 x^{2} - 2 x y + 4 y^{2} - z w - w^{2} $
	$=$	$2 x^{3} + 2 x^{2} y - x z^{2} - 3 x z w - x w^{2} + y z^{2} + y z w$

Singular plane model Singular plane model

$ 0 $

$=$

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

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

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle 2^4\,\frac{4502xyz^{8}+8646xyz^{7}w-27880xyz^{6}w^{2}-75388xyz^{5}w^{3}+28730xyz^{4}w^{4}+239356xyz^{3}w^{5}+227864xyz^{2}w^{6}+65104xyzw^{7}+2286y^{2}z^{8}+15382y^{2}z^{7}w+6608y^{2}z^{6}w^{2}-172044y^{2}z^{5}w^{3}-443290y^{2}z^{4}w^{4}-326254y^{2}z^{3}w^{5}+121028y^{2}z^{2}w^{6}+227864y^{2}zw^{7}+65104y^{2}w^{8}-1024z^{10}-6817z^{9}w-18947z^{8}w^{2}-18464z^{7}w^{3}+42554z^{6}w^{4}+150188z^{5}w^{5}+147652z^{4}w^{6}-14656z^{3}w^{7}-126544z^{2}w^{8}-81920zw^{9}-16384w^{10}}{14xyz^{8}+50xyz^{7}w+82xyz^{6}w^{2}+106xyz^{5}w^{3}+70xyz^{4}w^{4}+14xyz^{3}w^{5}-14xyz^{2}w^{6}-4xyzw^{7}+6y^{2}z^{8}+50y^{2}z^{7}w+66y^{2}z^{6}w^{2}+10y^{2}z^{5}w^{3}-10y^{2}z^{4}w^{4}-26y^{2}z^{3}w^{5}-18y^{2}z^{2}w^{6}-14y^{2}zw^{7}-4y^{2}w^{8}-5z^{9}w-29z^{8}w^{2}-59z^{7}w^{3}-53z^{6}w^{4}-17z^{5}w^{5}+13z^{4}w^{6}+16z^{3}w^{7}+4z^{2}w^{8}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.60.4.a.1 :

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

Equation of the image curve:

$0$

$=$

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

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank
$X_{S_4}(5)$	$5$	$24$	$12$	$0$	$0$
24.24.0-8.a.1.1	$24$	$5$	$5$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
24.24.0-8.a.1.1	$24$	$5$	$5$	$0$	$0$
60.60.2-10.a.1.1	$60$	$2$	$2$	$2$	$0$
120.60.2-10.a.1.1	$120$	$2$	$2$	$2$	$?$

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

Covering curve	Level	Index	Degree	Genus
120.240.8-40.a.1.1	$120$	$2$	$2$	$8$
120.240.8-40.a.1.2	$120$	$2$	$2$	$8$
120.240.8-40.c.1.2	$120$	$2$	$2$	$8$
120.240.8-40.c.1.7	$120$	$2$	$2$	$8$
120.240.8-40.e.1.2	$120$	$2$	$2$	$8$
120.240.8-40.e.1.5	$120$	$2$	$2$	$8$
120.240.8-40.g.1.2	$120$	$2$	$2$	$8$
120.240.8-40.g.1.4	$120$	$2$	$2$	$8$
120.360.10-40.a.1.3	$120$	$3$	$3$	$10$
120.480.13-40.y.1.3	$120$	$4$	$4$	$13$
120.240.8-120.b.1.1	$120$	$2$	$2$	$8$
120.240.8-120.b.1.7	$120$	$2$	$2$	$8$
120.240.8-120.d.1.2	$120$	$2$	$2$	$8$
120.240.8-120.d.1.14	$120$	$2$	$2$	$8$
120.240.8-120.i.1.2	$120$	$2$	$2$	$8$
120.240.8-120.i.1.9	$120$	$2$	$2$	$8$
120.240.8-120.k.1.5	$120$	$2$	$2$	$8$
120.240.8-120.k.1.6	$120$	$2$	$2$	$8$
120.360.14-120.a.1.16	$120$	$3$	$3$	$14$
120.480.17-120.fg.1.8	$120$	$4$	$4$	$17$