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Properties

Label 120.144.3-60.a.1.16
Level $120$
Index $144$
Genus $3$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $120$ $\SL_2$-level: $20$ Newform level: $720$
Index: $144$ $\PSL_2$-index:$72$
Genus: $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2 \le \gamma \le 3$
$\overline{\Q}$-gonality: $2 \le \gamma \le 3$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20J3

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}21&10\\50&59\end{bmatrix}$, $\begin{bmatrix}27&10\\110&61\end{bmatrix}$, $\begin{bmatrix}59&50\\40&9\end{bmatrix}$, $\begin{bmatrix}69&50\\28&7\end{bmatrix}$, $\begin{bmatrix}73&50\\86&69\end{bmatrix}$, $\begin{bmatrix}119&100\\68&51\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.72.3.a.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree: $16$
Cyclic 120-torsion field degree: $512$
Full 120-torsion field degree: $245760$

Models

Embedded model Embedded model in $\mathbb{P}^{4}$

$ 0 $ $=$ $ x y w - x y t - y^{2} w + y z w + w^{3} $
$=$ $x y w + 2 x y t - y^{2} w + y z w + w^{3} + w^{2} t$
$=$ $2 x y w + x y t + y^{2} w - y z w$
$=$ $3 x y z + z w^{2}$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} y + 4 x^{4} z - 3 x^{2} y^{2} z - 12 x^{2} y z^{2} - 15 x^{2} z^{3} + 9 y z^{4} + 36 z^{5} $
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Weierstrass model Weierstrass model

$ y^{2} + \left(x^{4} + 1\right) y $ $=$ $ 6x^{6} - 5x^{4} + 54x^{2} + 20 $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2}\cdot\frac{407558946816xz^{10}+305713004544xz^{8}t^{2}-26163689472xz^{6}t^{4}-231011136576xz^{4}t^{6}-676002211626xz^{2}t^{8}-3971447708xt^{10}-204862390272yz^{10}-194694119424yz^{8}t^{2}-10236284928yz^{6}t^{4}+119258296488yz^{4}t^{6}+360584539353yz^{2}t^{8}-17129878517760yw^{10}-27400266185728yw^{9}t+27694542697856yw^{8}t^{2}+83921870872064yw^{7}t^{3}+45630192510376yw^{6}t^{4}-33028680671924yw^{5}t^{5}-60985791011802yw^{4}t^{6}-39770355462912yw^{3}t^{7}-12954615914530yw^{2}t^{8}-1814799556016ywt^{9}+1677721600yt^{10}+203843174400z^{11}+195392176128z^{9}t^{2}+8425783296z^{7}t^{4}-119714715648z^{5}t^{6}-360504998208z^{3}t^{8}+1362870120448zw^{10}-16546947734528zw^{9}t-31467088636160zw^{8}t^{2}+23734136133696zw^{7}t^{3}+82535186182128zw^{6}t^{4}+43715663371140zw^{5}t^{5}-31113514820082zw^{4}t^{6}-52210715911103zw^{3}t^{7}-30344372768992zw^{2}t^{8}-8290652647973zwt^{9}-912963801354zt^{10}}{442368xz^{6}t^{4}-354816xz^{4}t^{6}+26304xz^{2}t^{8}-578789xt^{10}+221184yz^{6}t^{4}+46080yz^{4}t^{6}-33732yz^{2}t^{8}+5772902400yw^{10}+1210982400yw^{9}t-3323289600yw^{8}t^{2}-2916633600yw^{7}t^{3}-720295680yw^{6}t^{4}-29527104yw^{5}t^{5}+9333392yw^{4}t^{6}-3471972yw^{3}t^{7}+1023474yw^{2}t^{8}-690533ywt^{9}-147456z^{5}t^{6}+44544z^{3}t^{8}-398131200zw^{10}+6104678400zw^{9}t+611020800zw^{8}t^{2}-3737548800zw^{7}t^{3}-2189229056zw^{6}t^{4}-382860288zw^{5}t^{5}-9504000zw^{4}t^{6}+3723776zw^{3}t^{7}-1780834zw^{2}t^{8}+399993zwt^{9}-55872zt^{10}}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.72.3.a.1 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}t$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}w$

Equation of the image curve:

$0$ $=$ $ X^{4}Y+4X^{4}Z-3X^{2}Y^{2}Z-12X^{2}YZ^{2}-15X^{2}Z^{3}+9YZ^{4}+36Z^{5} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 60.72.3.a.1 :

$\displaystyle X$ $=$ $\displaystyle -w$
$\displaystyle Y$ $=$ $\displaystyle 4y^{4}-6y^{2}w^{2}-3y^{2}wt$
$\displaystyle Z$ $=$ $\displaystyle y$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
40.72.1-10.a.1.1 $40$ $2$ $2$ $1$ $0$
120.24.0-12.a.1.3 $120$ $6$ $6$ $0$ $?$
120.72.1-10.a.1.5 $120$ $2$ $2$ $1$ $?$

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

Covering curve Level Index Degree Genus
120.288.5-60.y.1.20 $120$ $2$ $2$ $5$
120.288.5-60.y.2.19 $120$ $2$ $2$ $5$
120.288.5-60.ba.1.6 $120$ $2$ $2$ $5$
120.288.5-60.ba.2.8 $120$ $2$ $2$ $5$
120.288.5-60.bg.1.6 $120$ $2$ $2$ $5$
120.288.5-60.bg.2.8 $120$ $2$ $2$ $5$
120.288.5-60.bi.1.8 $120$ $2$ $2$ $5$
120.288.5-60.bi.2.7 $120$ $2$ $2$ $5$
120.288.5-120.cv.1.9 $120$ $2$ $2$ $5$
120.288.5-120.cv.2.11 $120$ $2$ $2$ $5$
120.288.5-120.db.1.15 $120$ $2$ $2$ $5$
120.288.5-120.db.2.9 $120$ $2$ $2$ $5$
120.288.5-120.dt.1.11 $120$ $2$ $2$ $5$
120.288.5-120.dt.2.15 $120$ $2$ $2$ $5$
120.288.5-120.dz.1.11 $120$ $2$ $2$ $5$
120.288.5-120.dz.2.15 $120$ $2$ $2$ $5$
120.288.7-60.p.1.3 $120$ $2$ $2$ $7$
120.288.7-60.r.1.16 $120$ $2$ $2$ $7$
120.288.7-60.r.1.62 $120$ $2$ $2$ $7$
120.288.7-60.bb.1.3 $120$ $2$ $2$ $7$
120.288.7-60.bb.1.16 $120$ $2$ $2$ $7$
120.288.7-60.bc.1.11 $120$ $2$ $2$ $7$
120.288.7-60.bc.1.16 $120$ $2$ $2$ $7$
120.288.7-120.cq.1.9 $120$ $2$ $2$ $7$
120.288.7-120.cq.1.18 $120$ $2$ $2$ $7$
120.288.7-120.cu.1.10 $120$ $2$ $2$ $7$
120.288.7-120.cu.1.17 $120$ $2$ $2$ $7$
120.288.7-120.el.1.1 $120$ $2$ $2$ $7$
120.288.7-120.el.1.26 $120$ $2$ $2$ $7$
120.288.7-60.em.1.5 $120$ $2$ $2$ $7$
120.288.7-60.em.1.16 $120$ $2$ $2$ $7$
120.288.7-60.em.2.9 $120$ $2$ $2$ $7$
120.288.7-60.em.2.16 $120$ $2$ $2$ $7$
120.288.7-60.en.1.5 $120$ $2$ $2$ $7$
120.288.7-60.en.1.16 $120$ $2$ $2$ $7$
120.288.7-60.en.2.9 $120$ $2$ $2$ $7$
120.288.7-60.en.2.16 $120$ $2$ $2$ $7$
120.288.7-120.eo.1.2 $120$ $2$ $2$ $7$
120.288.7-120.eo.1.25 $120$ $2$ $2$ $7$
120.288.7-120.bcu.1.17 $120$ $2$ $2$ $7$
120.288.7-120.bcu.1.31 $120$ $2$ $2$ $7$
120.288.7-120.bcu.2.17 $120$ $2$ $2$ $7$
120.288.7-120.bcu.2.31 $120$ $2$ $2$ $7$
120.288.7-120.bcx.1.23 $120$ $2$ $2$ $7$
120.288.7-120.bcx.1.25 $120$ $2$ $2$ $7$
120.288.7-120.bcx.2.23 $120$ $2$ $2$ $7$
120.288.7-120.bcx.2.25 $120$ $2$ $2$ $7$
120.432.15-60.c.1.28 $120$ $3$ $3$ $15$