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Properties

Label 132.144.4-44.a.1.2
Level $132$
Index $144$
Genus $4$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $132$ $\SL_2$-level: $44$ Newform level: $176$
Index: $144$ $\PSL_2$-index:$72$
Genus: $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $2^{3}\cdot22^{3}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3 \le \gamma \le 4$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 22A4

Level structure

$\GL_2(\Z/132\Z)$-generators: $\begin{bmatrix}10&1\\49&36\end{bmatrix}$, $\begin{bmatrix}25&110\\4&117\end{bmatrix}$, $\begin{bmatrix}47&70\\72&115\end{bmatrix}$, $\begin{bmatrix}124&65\\97&112\end{bmatrix}$, $\begin{bmatrix}128&91\\51&110\end{bmatrix}$
Contains $-I$: no $\quad$ (see 44.72.4.a.1 for the level structure with $-I$)
Cyclic 132-isogeny field degree: $8$
Cyclic 132-torsion field degree: $320$
Full 132-torsion field degree: $422400$

Models

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

$ 0 $ $=$ $ 11 x^{2} + 2 x z + 4 y^{2} + 3 z^{2} - w^{2} $
$=$ $x^{3} - 3 x^{2} z + x y^{2} - x z^{2} - y^{2} z - z^{3}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{6} - 3 x^{4} y^{2} + 4 x^{4} z^{2} + x^{2} y^{4} + x^{2} z^{4} - 2 x y^{4} z + 2 x y^{2} z^{3} + \cdots + y^{2} z^{4} $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:-1/2:0:1)$, $(0:1/2:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2}\cdot\frac{656768101062205440xyz^{9}w-964981447482286080xyz^{7}w^{3}+154095616793341440xyz^{5}w^{5}+189018366871834080xyz^{3}w^{7}-2392922623412160xyzw^{9}-68917158293372928xz^{11}+1408109798333196288xz^{9}w^{2}-1104468169497724416xz^{7}w^{4}-681702629492929408xz^{5}w^{6}-92435619755648764xz^{3}w^{8}+10097890221697656xzw^{10}+426482095491809280y^{3}z^{8}w+1691502027435601920y^{3}z^{6}w^{3}-236208180084710400y^{3}z^{4}w^{5}-11916562162216320y^{3}z^{2}w^{7}+4055922088669080y^{3}w^{9}+181577584000237568y^{2}z^{10}+92669783736893440y^{2}z^{8}w^{2}+2037860252634187520y^{2}z^{6}w^{4}+198846456598237952y^{2}z^{4}w^{6}-27370078840568576y^{2}z^{2}w^{8}-1979403968349283y^{2}w^{10}+784670392474214400yz^{10}w-158541063406387200yz^{8}w^{3}+473418375973724160yz^{6}w^{5}+115754678398951200yz^{4}w^{7}+32607269469220320yz^{2}w^{9}+362179571158440yw^{11}+81634449242570752z^{12}+1116769383125817344z^{10}w^{2}+489577565635045888z^{8}w^{4}+226059822025657536z^{6}w^{6}-155209579182910308z^{4}w^{8}-11171867782966604z^{2}w^{10}-193229831382469w^{12}}{119248183311488xyz^{9}w-69460050972928xyz^{7}w^{3}-3297763465192xyz^{5}w^{5}+3137598103248xyz^{3}w^{7}-706778182656xyzw^{9}-234217785776128xz^{11}+266931634805440xz^{9}w^{2}-19759808546016xz^{7}w^{4}-25655941196340xz^{5}w^{6}+3406978923880xz^{3}w^{8}+873258989568xzw^{10}-15251031803392y^{3}z^{8}w+41492523616544y^{3}z^{6}w^{3}-11148812005184y^{3}z^{4}w^{5}+36619931950y^{3}z^{2}w^{7}-81921351680y^{3}w^{9}+14550315839488y^{2}z^{10}-139914140593920y^{2}z^{8}w^{2}+102512011752304y^{2}z^{6}w^{4}-30961546285656y^{2}z^{4}w^{6}+4529621933207y^{2}z^{2}w^{8}+33766742016y^{2}w^{10}+13763366207360yz^{10}w+49146944906752yz^{8}w^{3}-29539392715896yz^{6}w^{5}+3438948362360yz^{4}w^{7}+371491406818yz^{2}w^{9}+20480337920yw^{11}-38876063345152z^{12}-112374478324416z^{10}w^{2}+149339392794592z^{8}w^{4}-55920107263356z^{6}w^{6}+11814575612980z^{4}w^{8}-1869820697871z^{2}w^{10}-8441685504w^{12}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 44.72.4.a.1 :

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

Equation of the image curve:

$0$ $=$ $ 4X^{6}-3X^{4}Y^{2}+4X^{4}Z^{2}+X^{2}Y^{4}+X^{2}Z^{4}-2XY^{4}Z+2XY^{2}Z^{3}+Y^{4}Z^{2}+Y^{2}Z^{4} $

Modular covers

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

Factor curve Level Index Degree Genus Rank
$X_0(11)$ $11$ $12$ $6$ $1$ $0$
12.12.0-4.a.1.1 $12$ $12$ $12$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
12.12.0-4.a.1.1 $12$ $12$ $12$ $0$ $0$

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

Covering curve Level Index Degree Genus
132.288.8-44.b.1.2 $132$ $2$ $2$ $8$
132.288.8-44.b.1.3 $132$ $2$ $2$ $8$
132.288.8-44.b.2.1 $132$ $2$ $2$ $8$
132.288.8-44.b.2.4 $132$ $2$ $2$ $8$
132.288.8-132.b.1.1 $132$ $2$ $2$ $8$
132.288.8-132.b.1.8 $132$ $2$ $2$ $8$
132.288.8-132.b.2.2 $132$ $2$ $2$ $8$
132.288.8-132.b.2.7 $132$ $2$ $2$ $8$
132.288.9-44.e.1.6 $132$ $2$ $2$ $9$
132.288.9-132.e.1.12 $132$ $2$ $2$ $9$
132.288.9-44.f.1.6 $132$ $2$ $2$ $9$
132.288.9-132.f.1.9 $132$ $2$ $2$ $9$
132.288.9-44.g.1.6 $132$ $2$ $2$ $9$
132.288.9-132.g.1.12 $132$ $2$ $2$ $9$
132.288.9-44.h.1.8 $132$ $2$ $2$ $9$
132.288.9-132.h.1.10 $132$ $2$ $2$ $9$
132.288.10-44.b.1.2 $132$ $2$ $2$ $10$
132.288.10-44.b.1.3 $132$ $2$ $2$ $10$
132.288.10-44.b.2.2 $132$ $2$ $2$ $10$
132.288.10-44.b.2.3 $132$ $2$ $2$ $10$
132.288.10-132.b.1.5 $132$ $2$ $2$ $10$
132.288.10-132.b.1.11 $132$ $2$ $2$ $10$
132.288.10-132.b.2.7 $132$ $2$ $2$ $10$
132.288.10-132.b.2.9 $132$ $2$ $2$ $10$
132.432.16-132.a.1.1 $132$ $3$ $3$ $16$
264.288.8-88.b.1.1 $264$ $2$ $2$ $8$
264.288.8-88.b.1.8 $264$ $2$ $2$ $8$
264.288.8-88.b.2.3 $264$ $2$ $2$ $8$
264.288.8-88.b.2.6 $264$ $2$ $2$ $8$
264.288.8-264.qd.1.5 $264$ $2$ $2$ $8$
264.288.8-264.qd.1.12 $264$ $2$ $2$ $8$
264.288.8-264.qd.2.6 $264$ $2$ $2$ $8$
264.288.8-264.qd.2.11 $264$ $2$ $2$ $8$
264.288.9-88.m.1.9 $264$ $2$ $2$ $9$
264.288.9-88.p.1.10 $264$ $2$ $2$ $9$
264.288.9-88.s.1.12 $264$ $2$ $2$ $9$
264.288.9-88.v.1.11 $264$ $2$ $2$ $9$
264.288.9-264.cro.1.23 $264$ $2$ $2$ $9$
264.288.9-264.ctb.1.18 $264$ $2$ $2$ $9$
264.288.9-264.cyc.1.17 $264$ $2$ $2$ $9$
264.288.9-264.cyf.1.24 $264$ $2$ $2$ $9$
264.288.10-88.b.1.1 $264$ $2$ $2$ $10$
264.288.10-88.b.1.8 $264$ $2$ $2$ $10$
264.288.10-88.b.2.1 $264$ $2$ $2$ $10$
264.288.10-88.b.2.8 $264$ $2$ $2$ $10$
264.288.10-264.f.1.11 $264$ $2$ $2$ $10$
264.288.10-264.f.1.21 $264$ $2$ $2$ $10$
264.288.10-264.f.2.3 $264$ $2$ $2$ $10$
264.288.10-264.f.2.29 $264$ $2$ $2$ $10$