Modular curve 72.72.2-18.d.1.1

• Label: 72.72.2-18.d.1.1
• Level: $72$
• Index: $72$
• Genus: $2$
• Cusps: $4$
• $\Q$-cusps: $4$

Invariants

Level:	$72$		$\SL_2$-level:	$36$		Newform level:	$54$
Index:	$72$		$\PSL_2$-index:	$36$
Genus:	$2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (all of which are rational)		Cusp widths	$3\cdot6\cdot9\cdot18$		Cusp orbits	$1^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

18D2

Level structure

$\GL_2(\Z/72\Z)$-generators:	$\begin{bmatrix}6&31\\37&60\end{bmatrix}$, $\begin{bmatrix}39&62\\16&35\end{bmatrix}$, $\begin{bmatrix}56&37\\15&16\end{bmatrix}$, $\begin{bmatrix}58&65\\57&32\end{bmatrix}$, $\begin{bmatrix}60&31\\65&16\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 18.36.2.d.1 for the level structure with $-I$)
Cyclic 72-isogeny field degree:	$12$
Cyclic 72-torsion field degree:	$288$
Full 72-torsion field degree:	$82944$

Models

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

$ 0 $	$=$	$ x y w - x z w - w^{3} $
	$=$	$x y z - x z^{2} - z w^{2}$
	$=$	$x y^{2} - x y z - y w^{2}$
	$=$	$x^{2} y - x^{2} z - x w^{2}$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{4} y + x^{3} z^{2} + 12 x^{2} y^{2} z + 7 x y z^{3} + z^{5} $

Weierstrass model Weierstrass model

$ y^{2} + \left(x^{3} + 1\right) y $

$=$

$ -9x^{3} $

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal 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	Plane model	Weierstrass model	Embedded model
27.a3	$-3$	$0$		$0.000$	$(-1:0:1)$	$(-1:-3:1)$	$(-1:-1:0:1)$
	no	$\infty$		$0.000$
36.a1	$-12$	$54000$	$= 2^{4} \cdot 3^{3} \cdot 5^{3}$	$10.897$	$(-1:1/2:1)$	$(-1:3:1)$	$(-1:0:1:1)$

Maps to other modular curves

$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 3^3\,\frac{192x^{7}w-192x^{4}w^{4}+327xz^{6}w+7248xz^{3}w^{4}-2688xw^{7}+53y^{2}z^{6}-4096y^{2}z^{3}w^{3}-2304y^{2}w^{6}-16yz^{7}-6424yz^{4}w^{3}-576yzw^{6}-21z^{8}-1908z^{5}w^{3}+7200z^{2}w^{6}}{w(12xz^{6}-21xz^{3}w^{3}+22y^{2}z^{3}w^{2}-8y^{2}w^{5}+37yz^{4}w^{2}-20yzw^{5}+21z^{5}w^{2}-30z^{2}w^{5})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 18.36.2.d.1 :

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

Equation of the image curve:

$0$

$=$

$ X^{4}Y+12X^{2}Y^{2}Z+X^{3}Z^{2}+7XYZ^{3}+Z^{5} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 18.36.2.d.1 :

$\displaystyle X$	$=$	$\displaystyle -x$
$\displaystyle Y$	$=$	$\displaystyle x^{3}+6xzw+4w^{3}$
$\displaystyle Z$	$=$	$\displaystyle -w$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
24.24.0-6.a.1.10	$24$	$3$	$3$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
72.144.4-18.b.1.3	$72$	$2$	$2$	$4$
72.144.4-36.b.1.3	$72$	$2$	$2$	$4$
72.144.4-72.b.1.2	$72$	$2$	$2$	$4$
72.144.4-72.d.1.8	$72$	$2$	$2$	$4$
72.144.4-36.e.1.11	$72$	$2$	$2$	$4$
72.144.4-36.f.1.27	$72$	$2$	$2$	$4$
72.144.4-72.g.1.15	$72$	$2$	$2$	$4$
72.144.4-72.h.1.9	$72$	$2$	$2$	$4$
72.144.4-18.l.1.3	$72$	$2$	$2$	$4$
72.144.4-36.m.1.2	$72$	$2$	$2$	$4$
72.144.4-36.p.1.1	$72$	$2$	$2$	$4$
72.144.4-36.q.1.1	$72$	$2$	$2$	$4$
72.144.4-72.r.1.12	$72$	$2$	$2$	$4$
72.144.4-72.t.1.12	$72$	$2$	$2$	$4$
72.144.4-72.w.1.1	$72$	$2$	$2$	$4$
72.144.4-72.x.1.3	$72$	$2$	$2$	$4$
72.144.5-36.l.1.6	$72$	$2$	$2$	$5$
72.144.5-36.m.1.6	$72$	$2$	$2$	$5$
72.144.5-36.n.1.4	$72$	$2$	$2$	$5$
72.144.5-36.o.1.18	$72$	$2$	$2$	$5$
72.144.5-72.be.1.12	$72$	$2$	$2$	$5$
72.144.5-72.bf.1.12	$72$	$2$	$2$	$5$
72.144.5-72.bg.1.8	$72$	$2$	$2$	$5$
72.144.5-72.bh.1.2	$72$	$2$	$2$	$5$
72.216.4-18.c.1.14	$72$	$3$	$3$	$4$
72.216.4-18.g.1.1	$72$	$3$	$3$	$4$
72.216.4-18.g.2.1	$72$	$3$	$3$	$4$
72.216.4-18.h.1.1	$72$	$3$	$3$	$4$
216.216.8-54.a.1.15	$216$	$3$	$3$	$8$
216.216.8-54.b.1.14	$216$	$3$	$3$	$8$
216.216.8-54.c.1.9	$216$	$3$	$3$	$8$