Modular curve 72.144.4-18.q.1.3

• Label: 72.144.4-18.q.1.3
• Level: $72$
• Index: $144$
• Genus: $4$
• Cusps: $6$
• $\Q$-cusps: $0$

Invariants

Level:	$72$		$\SL_2$-level:	$18$		Newform level:	$324$
Index:	$144$		$\PSL_2$-index:	$72$
Genus:	$4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (none of which are rational)		Cusp widths	$6^{3}\cdot18^{3}$		Cusp orbits	$3^{2}$
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:

18I4

Level structure

$\GL_2(\Z/72\Z)$-generators:	$\begin{bmatrix}63&32\\5&63\end{bmatrix}$, $\begin{bmatrix}64&33\\1&44\end{bmatrix}$, $\begin{bmatrix}71&45\\14&55\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 18.72.4.q.1 for the level structure with $-I$)
Cyclic 72-isogeny field degree:	$36$
Cyclic 72-torsion field degree:	$864$
Full 72-torsion field degree:	$41472$

Models

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

$ 0 $	$=$	$ 9 x^{2} - y^{2} - y z - z^{2} + w^{2} $
	$=$	$3 y^{2} z - 2 y^{2} w + 3 y z^{2} - 2 y z w - 2 z^{2} w + w^{3}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 9 x^{4} y^{2} - 12 x^{4} y z + 4 x^{4} z^{2} + 54 x^{3} y^{3} - 72 x^{3} y^{2} z + 24 x^{3} y z^{2} + \cdots + 100 z^{6} $

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve	CM	$j$-invariant		$j$-height	Canonical model	Plane model
27.a3	$-3$	$0$		$0.000$	$(-1/3:-1:1:0)$, $(1/3:-1:1:0)$, $(-1/3:0:1:0)$, $(1/3:0:1:0)$, $(-1/3:1:0:0)$, $(1/3:1:0:0)$	$(-2:1:0)$, $(-4:1:0)$, $(1:0:0)$, $(-1:1:0)$, $(1:1:0)$

Maps to other modular curves

$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 3^7\,\frac{w^{3}(124416y^{9}+373248y^{8}w+124416y^{7}w^{2}-737856y^{6}w^{3}-770688y^{5}w^{4}+342720y^{4}w^{5}+790088y^{3}w^{6}+116168y^{2}w^{7}-373248yz^{8}-622080yz^{7}w+456192yz^{6}w^{2}+644544yz^{5}w^{3}-441216yz^{4}w^{4}-101184yz^{3}w^{5}-330840yz^{2}w^{6}+116936yzw^{7}-145024yw^{8}-124416z^{9}-124416z^{8}w+290304z^{7}w^{2}-46656z^{6}w^{3}-605952z^{5}w^{4}+120768z^{4}w^{5}+229624z^{3}w^{6}+116552z^{2}w^{7}-72512zw^{8}-134395w^{9})}{19683y^{12}+78732y^{11}w+65610y^{10}w^{2}-86751y^{9}w^{3}-100602y^{8}w^{4}+64881y^{7}w^{5}+49653y^{6}w^{6}-43335y^{5}w^{7}-3510y^{4}w^{8}+11714y^{3}w^{9}-4972y^{2}w^{10}+19683yz^{11}+52488yz^{10}w-4374yz^{9}w^{2}-158193yz^{8}w^{3}-122958yz^{7}w^{4}+149121yz^{6}w^{5}+161703yz^{5}w^{6}-70803yz^{4}w^{7}-70422yz^{3}w^{8}+21996yz^{2}w^{9}+7640yzw^{10}-1402yw^{11}+19683z^{12}+65610z^{11}w+30618z^{10}w^{2}-122472z^{9}w^{3}-111780z^{8}w^{4}+107001z^{7}w^{5}+105678z^{6}w^{6}-57069z^{5}w^{7}-36966z^{4}w^{8}+16855z^{3}w^{9}+1334z^{2}w^{10}-701zw^{11}+164w^{12}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 18.72.4.q.1 :

$\displaystyle X$	$=$	$\displaystyle x+z$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{3}y$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{3}w$

Equation of the image curve:

$0$

$=$

$ 9X^{4}Y^{2}-12X^{4}YZ+4X^{4}Z^{2}+54X^{3}Y^{3}-72X^{3}Y^{2}Z+24X^{3}YZ^{2}+63X^{2}Y^{4}-204X^{2}Y^{3}Z+126X^{2}Y^{2}Z^{2}+36X^{2}YZ^{3}-32X^{2}Z^{4}-54XY^{5}-288XY^{4}Z+270XY^{3}Z^{2}+108XY^{2}Z^{3}-96XYZ^{4}-72Y^{6}-54Y^{5}Z+387Y^{4}Z^{2}+87Y^{3}Z^{3}-333Y^{2}Z^{4}-60YZ^{5}+100Z^{6} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
72.48.2-18.b.1.2	$72$	$3$	$3$	$2$	$?$
72.72.1-9.c.1.4	$72$	$2$	$2$	$1$	$?$
72.72.1-9.c.1.6	$72$	$2$	$2$	$1$	$?$

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

Covering curve	Level	Index	Degree	Genus
72.432.10-18.v.1.6	$72$	$3$	$3$	$10$
72.432.10-18.y.1.4	$72$	$3$	$3$	$10$
72.432.10-18.z.1.3	$72$	$3$	$3$	$10$
216.432.16-54.bf.1.1	$216$	$3$	$3$	$16$
216.432.16-54.bg.1.1	$216$	$3$	$3$	$16$