Modular curve 120.120.4-60.i.1.6

• Label: 120.120.4-60.i.1.6
• Level: $120$
• Index: $120$
• Genus: $4$
• Cusps: $4$
• $\Q$-cusps: $0$

Invariants

Level:	$120$		$\SL_2$-level:	$40$		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:	$4$ (none of which are rational)		Cusp widths	$10^{2}\cdot20^{2}$		Cusp orbits	$2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 6$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 4$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

20A4

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}39&52\\50&41\end{bmatrix}$, $\begin{bmatrix}73&99\\12&17\end{bmatrix}$, $\begin{bmatrix}79&33\\104&71\end{bmatrix}$, $\begin{bmatrix}79&84\\40&103\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.60.4.i.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 $	$=$	$ 5 x^{2} - x z - y^{2} + y w - z^{2} - w^{2} $
	$=$	$2 x^{3} + 2 x^{2} z + x y^{2} + x y w + 2 x z^{2} - y^{2} z + 2 y z w$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{6} + 2 x^{5} z + x^{4} y^{2} + 3 x^{4} z^{2} - 13 x^{3} y^{2} z + 2 x^{3} z^{3} + 12 x^{2} y^{4} + \cdots + 4 y^{2} z^{4} $

Rational points

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

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle -\frac{2^8}{3^4}\cdot\frac{7056920869756xyz^{7}w+22590700999755xyz^{5}w^{3}+8053613322251xyz^{3}w^{5}+683546236100xyzw^{7}+3102302699052xz^{9}+8860558214101xz^{7}w^{2}+97259667540xz^{5}w^{4}+557326406567xz^{3}w^{6}+87666961844xzw^{8}-1885611963739y^{3}z^{6}w+690154288355y^{3}z^{4}w^{3}+375423279126y^{3}z^{2}w^{5}-48260484440y^{3}w^{7}-301319260408y^{2}z^{8}+10036097064630y^{2}z^{6}w^{2}+9222456907930y^{2}z^{4}w^{4}+1912205456476y^{2}z^{2}w^{6}+79963939104y^{2}w^{8}+1919286091659yz^{8}w-1252639017630yz^{6}w^{3}-3744394428369yz^{4}w^{5}-81995431056yz^{2}w^{7}+32103792336yw^{9}+749838615732z^{10}+5560895410423z^{8}w^{2}+6403679816156z^{6}w^{4}+2002942334827z^{4}w^{6}+434643822746z^{2}w^{8}+31703454664w^{10}}{329695088xyz^{7}w-277580800xyz^{5}w^{3}-203506303xyz^{3}w^{5}-10204411xyzw^{7}-19197888xz^{9}-1097151616xz^{7}w^{2}-869508220xz^{5}w^{4}-132979273xz^{3}w^{6}-2510452xzw^{8}-28340032y^{3}z^{6}w-133213360y^{3}z^{4}w^{3}-22969828y^{3}z^{2}w^{5}-231511y^{3}w^{7}-72435648y^{2}z^{8}-31756144y^{2}z^{6}w^{2}+181835780y^{2}z^{4}w^{4}+25476500y^{2}z^{2}w^{6}+251594y^{2}w^{8}+175264816yz^{8}w-45762976yz^{6}w^{3}-190949563yz^{4}w^{5}-24234616yz^{2}w^{7}-251594yw^{9}-30582528z^{10}-347703584z^{8}w^{2}-470092336z^{6}w^{4}-161810488z^{4}w^{6}-8819125z^{2}w^{8}+20083w^{10}}$

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

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

Equation of the image curve:

$0$

$=$

$ X^{6}+2X^{5}Z+X^{4}Y^{2}+3X^{4}Z^{2}-13X^{3}Y^{2}Z+2X^{3}Z^{3}+12X^{2}Y^{4}-9X^{2}Y^{2}Z^{2}+X^{2}Z^{4}+12XY^{4}Z+8XY^{2}Z^{3}+12Y^{4}Z^{2}+4Y^{2}Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.60.2-20.b.1.6	$40$	$2$	$2$	$2$	$0$
120.24.0-12.e.1.1	$120$	$5$	$5$	$0$	$?$
120.60.2-20.b.1.5	$120$	$2$	$2$	$2$	$?$

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

Covering curve	Level	Index	Degree	Genus
120.360.10-60.q.1.5	$120$	$3$	$3$	$10$
120.360.14-60.bg.1.2	$120$	$3$	$3$	$14$
120.480.13-60.ea.1.7	$120$	$4$	$4$	$13$
120.480.17-60.q.1.15	$120$	$4$	$4$	$17$