Modular curve 72.72.1-9.b.1.2

• Label: 72.72.1-9.b.1.2
• Level: $72$
• Index: $72$
• Genus: $1$
• Cusps: $6$
• $\Q$-cusps: $3$

Invariants

Level:	$72$		$\SL_2$-level:	$9$		Newform level:	$27$
Index:	$72$		$\PSL_2$-index:	$36$
Genus:	$1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (of which $3$ are rational)		Cusp widths	$3^{3}\cdot9^{3}$		Cusp orbits	$1^{3}\cdot3$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$3$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

9D1

Level structure

$\GL_2(\Z/72\Z)$-generators:	$\begin{bmatrix}6&5\\7&2\end{bmatrix}$, $\begin{bmatrix}11&12\\45&35\end{bmatrix}$, $\begin{bmatrix}13&44\\60&23\end{bmatrix}$, $\begin{bmatrix}16&51\\51&70\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 9.36.1.b.1 for the level structure with $-I$)
Cyclic 72-isogeny field degree:	$36$
Cyclic 72-torsion field degree:	$288$
Full 72-torsion field degree:	$82944$

Jacobian

Conductor:	$?$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	27.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} + y $

$=$

$ x^{3} - 30x + 63 $

Rational points

This modular curve is an elliptic curve, but the rank has not been computed

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle -\frac{45x^{2}y^{13}+495x^{2}y^{12}z-282825x^{2}y^{11}z^{2}-2844144x^{2}y^{10}z^{3}+327288510x^{2}y^{9}z^{4}+3101227884x^{2}y^{8}z^{5}-143410415814x^{2}y^{7}z^{6}-1309723050969x^{2}y^{6}z^{7}+29020278578499x^{2}y^{5}z^{8}+259651549790595x^{2}y^{4}z^{9}-2748947625186660x^{2}y^{3}z^{10}-24384908861977626x^{2}y^{2}z^{11}+98838544660227945x^{2}yz^{12}+876340322794896534x^{2}z^{13}-1125xy^{13}z-12348xy^{12}z^{2}+3608361xy^{11}z^{3}+36670662xy^{10}z^{4}-3138258942xy^{9}z^{5}-30292328646xy^{8}z^{6}+1161497639160xy^{7}z^{7}+10838328285024xy^{6}z^{8}-209603521903275xy^{5}z^{9}-1915611990872508xy^{4}z^{10}+18255011879875725xy^{3}z^{11}+165027817631563227xy^{2}z^{12}-615091687170963948xyz^{13}-5540070494597252697xz^{14}-y^{15}-12y^{14}z+19461y^{13}z^{2}+213998y^{12}z^{3}-33331353y^{11}z^{4}-345250389y^{10}z^{5}+18983578571y^{9}z^{6}+190282570776y^{8}z^{7}-4901397940413y^{7}z^{8}-48411803440853y^{6}z^{9}+627287585677707y^{5}z^{10}+6185154738294402y^{4}z^{11}-38248946758108490y^{3}z^{12}-380343893183967099y^{2}z^{13}+861718640294882523yz^{14}+8733148578637689286z^{15}}{z^{3}(45x^{2}y^{10}+225x^{2}y^{9}z-262089x^{2}y^{8}z^{2}-1049706x^{2}y^{7}z^{3}+263419236x^{2}y^{6}z^{4}+793932624x^{2}y^{5}z^{5}-93588211881x^{2}y^{4}z^{6}-188500869909x^{2}y^{3}z^{7}+13682660170014x^{2}y^{2}z^{8}+13777043102091x^{2}yz^{9}-705919229267499x^{2}z^{10}-1107xy^{10}z-5535xy^{9}z^{2}+3229002xy^{8}z^{3}+12949218xy^{7}z^{4}-2421747522xy^{6}z^{5}-7310588076xy^{5}z^{6}+724178167614xy^{4}z^{7}+1460555767179xy^{3}z^{8}-94217145984351xy^{2}z^{9}-94948644459150xyz^{10}+4462698100163787xz^{11}-y^{12}-6y^{11}z+18639y^{10}z^{2}+93250y^{9}z^{3}-28253550y^{8}z^{4}-113573766y^{7}z^{5}+13449257812y^{6}z^{6}+40745673333y^{5}z^{7}-2691784943988y^{4}z^{8}-5451612032791y^{3}z^{9}+233847504392670y^{2}z^{10}+236580120311562yz^{11}-7034821237083870z^{12})}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
72.24.1-9.a.1.4	$72$	$3$	$3$	$1$	$?$	dimension zero

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
72.144.4-18.g.1.3	$72$	$2$	$2$	$4$	$?$	not computed
72.144.4-36.i.1.2	$72$	$2$	$2$	$4$	$?$	not computed
72.144.4-18.j.1.2	$72$	$2$	$2$	$4$	$?$	not computed
72.144.4-36.k.1.2	$72$	$2$	$2$	$4$	$?$	not computed
72.144.4-72.k.1.6	$72$	$2$	$2$	$4$	$?$	not computed
72.144.4-72.l.1.2	$72$	$2$	$2$	$4$	$?$	not computed
72.144.4-72.o.1.5	$72$	$2$	$2$	$4$	$?$	not computed
72.144.4-72.p.1.1	$72$	$2$	$2$	$4$	$?$	not computed
72.216.1-9.a.2.2	$72$	$3$	$3$	$1$	$?$	dimension zero
72.216.4-18.e.1.8	$72$	$3$	$3$	$4$	$?$	not computed
72.288.10-36.x.1.8	$72$	$4$	$4$	$10$	$?$	not computed