Modular curve 120.288.5-60.dx.1.4

• Label: 120.288.5-60.dx.1.4
• Level: $120$
• Index: $288$
• Genus: $5$
• Cusps: $16$
• $\Q$-cusps: $4$

Invariants

Level:	$120$		$\SL_2$-level:	$12$		Newform level:	$3600$
Index:	$288$		$\PSL_2$-index:	$144$
Genus:	$5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (of which $4$ are rational)		Cusp widths	$6^{8}\cdot12^{8}$		Cusp orbits	$1^{4}\cdot2^{4}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 5$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 5$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

12B5

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}31&40\\81&41\end{bmatrix}$, $\begin{bmatrix}37&108\\84&7\end{bmatrix}$, $\begin{bmatrix}41&44\\12&49\end{bmatrix}$, $\begin{bmatrix}47&24\\39&41\end{bmatrix}$, $\begin{bmatrix}49&36\\15&37\end{bmatrix}$, $\begin{bmatrix}59&68\\69&115\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 60.144.5.dx.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree:	$12$
Cyclic 120-torsion field degree:	$384$
Full 120-torsion field degree:	$122880$

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $	$=$	$ 3 y^{2} - z^{2} + z w + z t + w t - t^{2} $
	$=$	$3 y^{2} - z w + w^{2} - w t$
	$=$	$5 x^{2} - y^{2} + z w - z t + w t$

Singular plane model Singular plane model

$ 0 $

$=$

$ 25 x^{4} z^{4} - 150 x^{3} y^{2} z^{3} - 50 x^{3} z^{5} + 345 x^{2} y^{4} z^{2} + 170 x^{2} y^{2} z^{4} + \cdots + 6 z^{8} $

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.

Canonical model

$(0:-1:3:1:1)$, $(0:1/3:1/3:1/3:1)$, $(0:1:3:1:1)$, $(0:-1/3:1/3:1/3:1)$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 3^3\,\frac{54zw^{17}+639zw^{16}t-144zw^{15}t^{2}-2880zw^{14}t^{3}+13004zw^{13}t^{4}-6578zw^{12}t^{5}-26160zw^{11}t^{6}+79276zw^{10}t^{7}-64150zw^{9}t^{8}-23067zw^{8}t^{9}+110664zw^{7}t^{10}-94248zw^{6}t^{11}+19638zw^{5}t^{12}+12627zw^{4}t^{13}-7848zw^{3}t^{14}+1566zw^{2}t^{15}-108zwt^{16}-45w^{18}-234w^{17}t+1188w^{16}t^{2}-1296w^{15}t^{3}-6182w^{14}t^{4}+23764w^{13}t^{5}-31994w^{12}t^{6}-5896w^{11}t^{7}+90221w^{10}t^{8}-147214w^{9}t^{9}+104364w^{8}t^{10}+6408w^{7}t^{11}-76563w^{6}t^{12}+63954w^{5}t^{13}-24399w^{4}t^{14}+3924w^{3}t^{15}+108w^{2}t^{16}-108wt^{17}+9t^{18}}{t^{3}w^{6}(w-t)^{3}(45zw^{5}+467zw^{4}t+889zw^{3}t^{2}+225zw^{2}t^{3}-90zwt^{4}-36w^{6}-170w^{5}t+196w^{4}t^{2}+271w^{3}t^{3}-261w^{2}t^{4}-9wt^{5}+9t^{6})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.5.dx.1 :

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

Equation of the image curve:

$0$

$=$

$ 25X^{4}Z^{4}-150X^{3}Y^{2}Z^{3}-50X^{3}Z^{5}+345X^{2}Y^{4}Z^{2}+170X^{2}Y^{2}Z^{4}+35X^{2}Z^{6}-360XY^{6}Z-180XY^{4}Z^{3}-50XY^{2}Z^{5}-10XZ^{7}+189Y^{8}-72Y^{6}Z^{2}+58Y^{4}Z^{4}+44Y^{2}Z^{6}+6Z^{8} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_{\mathrm{sp}}(3)$	$3$	$24$	$12$	$0$	$0$
40.24.0-20.h.1.2	$40$	$12$	$12$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
24.144.1-12.f.1.16	$24$	$2$	$2$	$1$	$0$
120.96.1-60.l.1.11	$120$	$3$	$3$	$1$	$?$
120.144.1-12.f.1.15	$120$	$2$	$2$	$1$	$?$
120.144.3-60.gc.1.2	$120$	$2$	$2$	$3$	$?$
120.144.3-60.gc.1.6	$120$	$2$	$2$	$3$	$?$
120.144.3-60.hy.1.1	$120$	$2$	$2$	$3$	$?$
120.144.3-60.hy.1.16	$120$	$2$	$2$	$3$	$?$