Modular curve 120.120.4-60.a.1.6

• Label: 120.120.4-60.a.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:	$\begin{bmatrix}9&16\\82&89\end{bmatrix}$, $\begin{bmatrix}35&106\\6&77\end{bmatrix}$, $\begin{bmatrix}53&50\\72&19\end{bmatrix}$, $\begin{bmatrix}77&16\\38&53\end{bmatrix}$, $\begin{bmatrix}93&26\\76&25\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.60.4.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 $	$=$	$ 6 x^{2} + 3 x y + 3 y^{2} - z w - w^{2} $
	$=$	$3 x y^{2} - x z w - 3 y^{3} - y z^{2} - y z w + y w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ - 108 x^{6} - 45 x^{4} y^{2} - 9 x^{4} y z + 54 x^{4} z^{2} - 6 x^{2} y^{4} - 9 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

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

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle 2^4\,\frac{3324xyz^{8}-10104xyz^{7}w-51732xyz^{6}w^{2}+144984xyz^{5}w^{3}+708030xyz^{4}w^{4}+848415xyz^{3}w^{5}+160254xyz^{2}w^{6}-244140xyzw^{7}-97656xyw^{8}-10182y^{2}z^{8}-36042y^{2}z^{7}w+31908y^{2}z^{6}w^{2}+371148y^{2}z^{5}w^{3}+621840y^{2}z^{4}w^{4}+130347y^{2}z^{3}w^{5}-523338y^{2}z^{2}w^{6}-439452y^{2}zw^{7}-97656y^{2}w^{8}-2048z^{10}-10240z^{9}w-22486z^{8}w^{2}-35550z^{7}w^{3}-49244z^{6}w^{4}-30620z^{5}w^{5}+44575z^{4}w^{6}+101685z^{3}w^{7}+67842z^{2}w^{8}+15196zw^{9}-216w^{10}}{12xyz^{8}+24xyz^{6}w^{2}+144xyz^{5}w^{3}+120xyz^{4}w^{4}+60xyz^{3}w^{5}+6xyz^{2}w^{6}+15xyzw^{7}+6xyw^{8}-30y^{2}z^{8}-150y^{2}z^{7}w-222y^{2}z^{6}w^{2}-174y^{2}z^{5}w^{3}-90y^{2}z^{4}w^{4}+18y^{2}z^{3}w^{5}+48y^{2}z^{2}w^{6}+27y^{2}zw^{7}+6y^{2}w^{8}+2z^{8}w^{2}+6z^{7}w^{3}+26z^{6}w^{4}+54z^{5}w^{5}+50z^{4}w^{6}+10z^{3}w^{7}-17z^{2}w^{8}-11zw^{9}-2w^{10}}$

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

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

Equation of the image curve:

$0$

$=$

$ -108X^{6}-45X^{4}Y^{2}-9X^{4}YZ+54X^{4}Z^{2}-6X^{2}Y^{4}-9X^{2}Y^{3}Z+3X^{2}YZ^{3}-6X^{2}Z^{4}+Y^{3}Z^{3}+Y^{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
$X_{S_4}(5)$	$5$	$24$	$12$	$0$	$0$
24.24.0-12.a.1.3	$24$	$5$	$5$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
24.24.0-12.a.1.3	$24$	$5$	$5$	$0$	$0$
40.60.2-10.a.1.1	$40$	$2$	$2$	$2$	$0$
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.a.1.2	$120$	$2$	$2$	$8$
120.240.8-120.b.1.6	$120$	$2$	$2$	$8$
120.240.8-120.b.1.9	$120$	$2$	$2$	$8$
120.240.8-60.c.1.8	$120$	$2$	$2$	$8$
120.240.8-60.c.1.16	$120$	$2$	$2$	$8$
120.240.8-120.f.1.1	$120$	$2$	$2$	$8$
120.240.8-120.f.1.14	$120$	$2$	$2$	$8$
120.240.8-60.g.1.2	$120$	$2$	$2$	$8$
120.240.8-60.g.1.8	$120$	$2$	$2$	$8$
120.240.8-60.h.1.6	$120$	$2$	$2$	$8$
120.240.8-60.h.1.8	$120$	$2$	$2$	$8$
120.240.8-120.o.1.3	$120$	$2$	$2$	$8$
120.240.8-120.o.1.16	$120$	$2$	$2$	$8$
120.240.8-120.r.1.8	$120$	$2$	$2$	$8$
120.240.8-120.r.1.11	$120$	$2$	$2$	$8$
120.360.10-60.a.1.12	$120$	$3$	$3$	$10$
120.360.14-60.c.1.2	$120$	$3$	$3$	$14$
120.480.13-60.y.1.4	$120$	$4$	$4$	$13$
120.480.17-60.c.1.16	$120$	$4$	$4$	$17$