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Properties

Label 120.72.2-60.s.1.14
Level $120$
Index $72$
Genus $2$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $120$ $\SL_2$-level: $24$ Newform level: $900$
Index: $72$ $\PSL_2$-index:$36$
Genus: $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $6^{2}\cdot12^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12B2

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}5&111\\48&113\end{bmatrix}$, $\begin{bmatrix}25&36\\24&79\end{bmatrix}$, $\begin{bmatrix}37&48\\16&107\end{bmatrix}$, $\begin{bmatrix}43&64\\60&77\end{bmatrix}$, $\begin{bmatrix}55&41\\8&47\end{bmatrix}$, $\begin{bmatrix}81&38\\56&17\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.36.2.s.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree: $48$
Cyclic 120-torsion field degree: $1536$
Full 120-torsion field degree: $491520$

Models

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

$ 0 $ $=$ $ 5 x^{2} w - y w^{2} + z w^{2} $
$=$ $5 x^{2} y - y^{2} w + y z w$
$=$ $5 x^{2} z - y z w + z^{2} w$
$=$ $5 x^{3} - x y w + x z w$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 10 x^{5} + 50 x^{3} y z - x^{2} z^{3} + 50 x y^{2} z^{2} - 2 y z^{4} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{6} + 125 $
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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.

Embedded model
$(0:0:0:1)$, $(0:1:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{47000xyz^{5}w-178320xyz^{2}w^{4}+27000xz^{6}w-148280xz^{3}w^{4}+256xw^{7}-1625y^{2}z^{6}+26630y^{2}z^{3}w^{3}-11072y^{2}w^{6}+1000yz^{7}-52110yz^{4}w^{3}+99520yzw^{6}+625z^{8}-64170z^{5}w^{3}+110592z^{2}w^{6}}{w^{6}(y-z)(y+z)}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.36.2.s.1 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{10}z$
$\displaystyle Z$ $=$ $\displaystyle 2w$

Equation of the image curve:

$0$ $=$ $ 10X^{5}+50X^{3}YZ+50XY^{2}Z^{2}-X^{2}Z^{3}-2YZ^{4} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 60.36.2.s.1 :

$\displaystyle X$ $=$ $\displaystyle w$
$\displaystyle Y$ $=$ $\displaystyle \frac{25}{8}x^{3}+\frac{5}{4}xzw-w^{3}$
$\displaystyle Z$ $=$ $\displaystyle -\frac{1}{2}x$

Modular covers

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

Factor curve Level Index Degree Genus Rank
$X_{\mathrm{ns}}^+(3)$ $3$ $24$ $12$ $0$ $0$
40.24.0-20.g.1.3 $40$ $3$ $3$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
24.36.1-12.c.1.13 $24$ $2$ $2$ $1$ $0$
40.24.0-20.g.1.3 $40$ $3$ $3$ $0$ $0$
120.36.1-12.c.1.16 $120$ $2$ $2$ $1$ $?$

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

Covering curve Level Index Degree Genus
120.144.3-60.fk.1.9 $120$ $2$ $2$ $3$
120.144.3-60.fl.1.6 $120$ $2$ $2$ $3$
120.144.3-60.ga.1.10 $120$ $2$ $2$ $3$
120.144.3-60.gb.1.7 $120$ $2$ $2$ $3$
120.144.3-60.gq.1.7 $120$ $2$ $2$ $3$
120.144.3-60.gr.1.7 $120$ $2$ $2$ $3$
120.144.3-60.hg.1.6 $120$ $2$ $2$ $3$
120.144.3-60.hh.1.7 $120$ $2$ $2$ $3$
120.144.3-120.bia.1.14 $120$ $2$ $2$ $3$
120.144.3-120.bih.1.10 $120$ $2$ $2$ $3$
120.144.3-120.bmi.1.16 $120$ $2$ $2$ $3$
120.144.3-120.bmp.1.16 $120$ $2$ $2$ $3$
120.144.3-120.bqq.1.16 $120$ $2$ $2$ $3$
120.144.3-120.bqx.1.16 $120$ $2$ $2$ $3$
120.144.3-120.buy.1.13 $120$ $2$ $2$ $3$
120.144.3-120.bvf.1.14 $120$ $2$ $2$ $3$
120.144.4-120.jg.1.17 $120$ $2$ $2$ $4$
120.144.4-120.jg.1.27 $120$ $2$ $2$ $4$
120.144.4-120.jh.1.35 $120$ $2$ $2$ $4$
120.144.4-120.jh.1.41 $120$ $2$ $2$ $4$
120.144.4-120.jw.1.21 $120$ $2$ $2$ $4$
120.144.4-120.jw.1.31 $120$ $2$ $2$ $4$
120.144.4-120.jx.1.22 $120$ $2$ $2$ $4$
120.144.4-120.jx.1.26 $120$ $2$ $2$ $4$
120.144.4-120.ke.1.18 $120$ $2$ $2$ $4$
120.144.4-120.ke.1.30 $120$ $2$ $2$ $4$
120.144.4-120.kf.1.22 $120$ $2$ $2$ $4$
120.144.4-120.kf.1.26 $120$ $2$ $2$ $4$
120.144.4-120.ki.1.20 $120$ $2$ $2$ $4$
120.144.4-120.ki.1.32 $120$ $2$ $2$ $4$
120.144.4-120.kj.1.24 $120$ $2$ $2$ $4$
120.144.4-120.kj.1.28 $120$ $2$ $2$ $4$
120.144.4-120.km.1.16 $120$ $2$ $2$ $4$
120.144.4-120.km.1.24 $120$ $2$ $2$ $4$
120.144.4-120.kn.1.8 $120$ $2$ $2$ $4$
120.144.4-120.kn.1.32 $120$ $2$ $2$ $4$
120.144.4-120.kq.1.12 $120$ $2$ $2$ $4$
120.144.4-120.kq.1.20 $120$ $2$ $2$ $4$
120.144.4-120.kr.1.4 $120$ $2$ $2$ $4$
120.144.4-120.kr.1.28 $120$ $2$ $2$ $4$
120.144.4-120.ku.1.12 $120$ $2$ $2$ $4$
120.144.4-120.ku.1.20 $120$ $2$ $2$ $4$
120.144.4-120.kv.1.10 $120$ $2$ $2$ $4$
120.144.4-120.kv.1.30 $120$ $2$ $2$ $4$
120.144.4-120.ky.1.6 $120$ $2$ $2$ $4$
120.144.4-120.ky.1.34 $120$ $2$ $2$ $4$
120.144.4-120.kz.1.2 $120$ $2$ $2$ $4$
120.144.4-120.kz.1.22 $120$ $2$ $2$ $4$
120.360.14-60.bm.1.16 $120$ $5$ $5$ $14$
120.432.15-60.cc.1.28 $120$ $6$ $6$ $15$