Modular curve 120.144.4-24.iw.1.1

• Label: 120.144.4-24.iw.1.1
• Level: $120$
• Index: $144$
• Genus: $4$
• Cusps: $6$
• $\Q$-cusps: $0$

Invariants

Level:	$120$		$\SL_2$-level:	$24$		Newform level:	$144$
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	$6^{4}\cdot24^{2}$		Cusp orbits	$2^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 6$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 4$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

24J4

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}28&45\\79&74\end{bmatrix}$, $\begin{bmatrix}36&23\\49&66\end{bmatrix}$, $\begin{bmatrix}38&93\\65&22\end{bmatrix}$, $\begin{bmatrix}83&88\\118&9\end{bmatrix}$, $\begin{bmatrix}112&93\\43&26\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 24.72.4.iw.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree:	$48$
Cyclic 120-torsion field degree:	$1536$
Full 120-torsion field degree:	$245760$

Models

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

$ 0 $	$=$	$ 7 x^{2} - 4 x y + 4 y^{2} + z^{2} + z w + w^{2} $
	$=$	$3 x^{3} + x z w + 2 x w^{2} - 2 y z w - y w^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 9 x^{6} + 16 x^{4} z^{2} - 32 x^{3} y z^{2} + 48 x^{2} y^{2} z^{2} + 3 x^{2} z^{4} - 32 x y^{3} z^{2} + \cdots + 3 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 72 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{2^6}{3}\cdot\frac{1152xyz^{10}+12960xyz^{9}w-20880xyz^{8}w^{2}-414720xyz^{7}w^{3}-240912xyz^{6}w^{4}+3297096xyz^{5}w^{5}+3155652xyz^{4}w^{6}-7387056xyz^{3}w^{7}-6525612xyz^{2}w^{8}+2661750xyzw^{9}+1126665xyw^{10}-1152y^{2}z^{10}-2880y^{2}z^{9}w+66240y^{2}z^{8}w^{2}+180864y^{2}z^{7}w^{3}-789264y^{2}z^{6}w^{4}-2007360y^{2}z^{5}w^{5}+2749968y^{2}z^{4}w^{6}+5984928y^{2}z^{3}w^{7}-2028528y^{2}z^{2}w^{8}-3438036y^{2}zw^{9}+133776y^{2}w^{10}-176z^{12}-336z^{11}w+7152z^{10}w^{2}+12896z^{9}w^{3}-81108z^{8}w^{4}-139860z^{7}w^{5}+292164z^{6}w^{6}+584424z^{5}w^{7}-43623z^{4}w^{8}-781963z^{3}w^{9}-867480z^{2}w^{10}-188169zw^{11}+194227w^{12}}{w^{2}(960xyz^{8}+8640xyz^{7}w+8544xyz^{6}w^{2}-34608xyz^{5}w^{3}-66168xyz^{4}w^{4}-30336xyz^{3}w^{5}+4164xyz^{2}w^{6}+4554xyzw^{7}+489xyw^{8}-960y^{2}z^{8}-1920y^{2}z^{7}w+14976y^{2}z^{6}w^{2}+39648y^{2}z^{5}w^{3}+19968y^{2}z^{4}w^{4}-14688y^{2}z^{3}w^{5}-13740y^{2}z^{2}w^{6}-2496y^{2}zw^{7}+36y^{2}w^{8}-128z^{10}-160z^{9}w+1260z^{8}w^{2}+2760z^{7}w^{3}-168z^{6}w^{4}-6408z^{5}w^{5}-9591z^{4}w^{6}-6555z^{3}w^{7}-1656z^{2}w^{8}+155zw^{9}+79w^{10})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.iw.1 :

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

Equation of the image curve:

$0$

$=$

$ 9X^{6}+16X^{4}Z^{2}-32X^{3}YZ^{2}+48X^{2}Y^{2}Z^{2}+3X^{2}Z^{4}-32XY^{3}Z^{2}-3XYZ^{4}+16Y^{4}Z^{2}+3Y^{2}Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
60.72.2-12.bb.1.4	$60$	$2$	$2$	$2$	$0$
120.72.2-12.bb.1.5	$120$	$2$	$2$	$2$	$?$
120.72.2-24.cs.1.7	$120$	$2$	$2$	$2$	$?$
120.72.2-24.cs.1.24	$120$	$2$	$2$	$2$	$?$
120.72.2-24.cw.1.14	$120$	$2$	$2$	$2$	$?$
120.72.2-24.cw.1.32	$120$	$2$	$2$	$2$	$?$

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

Covering curve	Level	Index	Degree	Genus
120.288.7-24.gz.1.2	$120$	$2$	$2$	$7$
120.288.7-24.jo.1.1	$120$	$2$	$2$	$7$
120.288.7-24.uj.1.1	$120$	$2$	$2$	$7$
120.288.7-24.un.1.1	$120$	$2$	$2$	$7$
120.288.7-24.zu.1.7	$120$	$2$	$2$	$7$
120.288.7-24.zy.1.5	$120$	$2$	$2$	$7$
120.288.7-24.bao.1.5	$120$	$2$	$2$	$7$
120.288.7-24.bas.1.5	$120$	$2$	$2$	$7$
120.288.7-120.fjl.1.1	$120$	$2$	$2$	$7$
120.288.7-120.fjp.1.1	$120$	$2$	$2$	$7$
120.288.7-120.fkr.1.5	$120$	$2$	$2$	$7$
120.288.7-120.fkv.1.1	$120$	$2$	$2$	$7$
120.288.7-120.foj.1.3	$120$	$2$	$2$	$7$
120.288.7-120.fon.1.10	$120$	$2$	$2$	$7$
120.288.7-120.fpp.1.9	$120$	$2$	$2$	$7$
120.288.7-120.fpt.1.9	$120$	$2$	$2$	$7$
240.288.9-48.gr.1.15	$240$	$2$	$2$	$9$
240.288.9-48.gt.1.16	$240$	$2$	$2$	$9$
240.288.9-48.lp.1.16	$240$	$2$	$2$	$9$
240.288.9-48.lr.1.15	$240$	$2$	$2$	$9$
240.288.9-48.qf.1.2	$240$	$2$	$2$	$9$
240.288.9-48.qh.1.1	$240$	$2$	$2$	$9$
240.288.9-48.rl.1.1	$240$	$2$	$2$	$9$
240.288.9-48.rn.1.2	$240$	$2$	$2$	$9$
240.288.9-240.bhl.1.6	$240$	$2$	$2$	$9$
240.288.9-240.bhn.1.2	$240$	$2$	$2$	$9$
240.288.9-240.bjh.1.20	$240$	$2$	$2$	$9$
240.288.9-240.bjj.1.24	$240$	$2$	$2$	$9$
240.288.9-240.bxp.1.17	$240$	$2$	$2$	$9$
240.288.9-240.bxr.1.25	$240$	$2$	$2$	$9$
240.288.9-240.byv.1.31	$240$	$2$	$2$	$9$
240.288.9-240.byx.1.27	$240$	$2$	$2$	$9$