↑

Properties

Label 120.144.3-60.ca.1.1
Level $120$
Index $144$
Genus $3$
Cusps $8$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $120$ $\SL_2$-level: $20$ Newform level: $360$
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: 20I3

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}1&90\\58&17\end{bmatrix}$, $\begin{bmatrix}29&70\\90&103\end{bmatrix}$, $\begin{bmatrix}37&100\\92&7\end{bmatrix}$, $\begin{bmatrix}47&80\\54&73\end{bmatrix}$, $\begin{bmatrix}77&50\\110&73\end{bmatrix}$, $\begin{bmatrix}107&100\\112&51\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.72.3.ca.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 z w - y z t $
$=$ $x y w - y^{2} t$
$=$ $x w t - y t^{2}$
$=$ $x w^{2} - y w t$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 5 x^{4} y + 5 x^{4} z - 3 x^{2} y^{2} z + 6 x^{2} y z^{2} + 6 x^{2} z^{3} + 9 y z^{4} + 9 z^{5} $
 Copy content Toggle raw display

Weierstrass model Weierstrass model

$ y^{2} + \left(x^{4} + 1\right) y $ $=$ $ 6x^{6} + 49x^{4} + 270x^{2} + 506 $
 Copy content Toggle raw display

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
$(0:0:0:0:1)$, $(1:0:0:0:0)$, $(1:-1:1:0:0)$, $(0:0:1:0:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{474609375x^{11}+506250000x^{9}t^{2}+141328125x^{7}t^{4}-16875000x^{5}t^{6}-8831250x^{3}t^{8}+1405000xt^{10}+474609375yz^{10}+348046875yz^{8}t^{2}+154828125yz^{6}t^{4}+27613125yz^{4}t^{6}+1527750yz^{2}t^{8}-3089408yw^{10}-19243008yw^{9}t-83523584yw^{8}t^{2}-197888768yw^{7}t^{3}-301738048yw^{6}t^{4}-295364864yw^{5}t^{5}-173919712yw^{4}t^{6}-33864400yw^{3}t^{7}+33598200yw^{2}t^{8}+32407312ywt^{9}+12433734yt^{10}+38812500z^{7}t^{4}+8010000z^{5}t^{6}+3810000z^{3}t^{8}-8385536zw^{10}-42356736zw^{9}t-132161536zw^{8}t^{2}-252640256zw^{7}t^{3}-309848768zw^{6}t^{4}-229691904zw^{5}t^{5}-64871168zw^{4}t^{6}+52239904zw^{3}t^{7}+69537720zw^{2}t^{8}+33348928zwt^{9}+7024984zt^{10}}{5625yz^{4}t^{6}+1792yw^{10}+11136yw^{9}t+25984yw^{8}t^{2}+31648yw^{7}t^{3}+22560yw^{6}t^{4}+9672yw^{5}t^{5}+2488yw^{4}t^{6}+310yw^{3}t^{7}-45yw^{2}t^{8}-20ywt^{9}+5625z^{5}t^{6}-375z^{3}t^{8}+4864zw^{10}+24512zw^{9}t+49920zw^{8}t^{2}+54160zw^{7}t^{3}+34192zw^{6}t^{4}+12724zw^{5}t^{5}+2728zw^{4}t^{6}+295zw^{3}t^{7}-50zw^{2}t^{8}+5zwt^{9}}$

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

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

Equation of the image curve:

$0$ $=$ $ 5X^{4}Y+5X^{4}Z-3X^{2}Y^{2}Z+6X^{2}YZ^{2}+6X^{2}Z^{3}+9YZ^{4}+9Z^{5} $

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

$\displaystyle X$ $=$ $\displaystyle -w$
$\displaystyle Y$ $=$ $\displaystyle 22y^{4}+3y^{2}w^{2}-3y^{2}wt$
$\displaystyle Z$ $=$ $\displaystyle y$

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 $40$ $2$ $2$ $1$ $0$
120.72.1-10.a.1.3 $120$ $2$ $2$ $1$ $?$

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.z.2.8 $120$ $2$ $2$ $5$
120.288.5-60.ba.1.1 $120$ $2$ $2$ $5$
120.288.5-60.bd.2.6 $120$ $2$ $2$ $5$
120.288.5-60.be.1.1 $120$ $2$ $2$ $5$
120.288.5-60.bl.1.8 $120$ $2$ $2$ $5$
120.288.5-60.bm.2.2 $120$ $2$ $2$ $5$
120.288.5-60.bp.1.1 $120$ $2$ $2$ $5$
120.288.5-60.bq.2.2 $120$ $2$ $2$ $5$
120.288.5-120.cz.2.2 $120$ $2$ $2$ $5$
120.288.5-120.dc.1.4 $120$ $2$ $2$ $5$
120.288.5-120.dl.2.4 $120$ $2$ $2$ $5$
120.288.5-120.do.1.2 $120$ $2$ $2$ $5$
120.288.5-120.ej.1.1 $120$ $2$ $2$ $5$
120.288.5-120.em.2.8 $120$ $2$ $2$ $5$
120.288.5-120.ev.1.2 $120$ $2$ $2$ $5$
120.288.5-120.ey.2.4 $120$ $2$ $2$ $5$
120.288.7-60.ej.2.2 $120$ $2$ $2$ $7$
120.288.7-60.ej.2.5 $120$ $2$ $2$ $7$
120.288.7-60.el.1.13 $120$ $2$ $2$ $7$
120.288.7-60.el.1.28 $120$ $2$ $2$ $7$
120.288.7-60.em.2.4 $120$ $2$ $2$ $7$
120.288.7-60.em.2.9 $120$ $2$ $2$ $7$
120.288.7-60.ep.1.5 $120$ $2$ $2$ $7$
120.288.7-60.ep.1.11 $120$ $2$ $2$ $7$
120.288.7-60.eq.1.5 $120$ $2$ $2$ $7$
120.288.7-60.eq.1.10 $120$ $2$ $2$ $7$
120.288.7-60.es.2.4 $120$ $2$ $2$ $7$
120.288.7-60.es.2.9 $120$ $2$ $2$ $7$
120.288.7-60.et.1.5 $120$ $2$ $2$ $7$
120.288.7-60.et.1.12 $120$ $2$ $2$ $7$
120.288.7-60.ew.2.3 $120$ $2$ $2$ $7$
120.288.7-60.ew.2.9 $120$ $2$ $2$ $7$
120.288.7-120.bco.1.8 $120$ $2$ $2$ $7$
120.288.7-120.bco.1.20 $120$ $2$ $2$ $7$
120.288.7-120.bcs.1.14 $120$ $2$ $2$ $7$
120.288.7-120.bcs.1.20 $120$ $2$ $2$ $7$
120.288.7-120.bcv.1.8 $120$ $2$ $2$ $7$
120.288.7-120.bcv.1.18 $120$ $2$ $2$ $7$
120.288.7-120.bde.1.16 $120$ $2$ $2$ $7$
120.288.7-120.bde.1.20 $120$ $2$ $2$ $7$
120.288.7-120.bdh.1.12 $120$ $2$ $2$ $7$
120.288.7-120.bdh.1.24 $120$ $2$ $2$ $7$
120.288.7-120.bdl.1.4 $120$ $2$ $2$ $7$
120.288.7-120.bdl.1.22 $120$ $2$ $2$ $7$
120.288.7-120.bdo.1.10 $120$ $2$ $2$ $7$
120.288.7-120.bdo.1.24 $120$ $2$ $2$ $7$
120.288.7-120.bdx.1.4 $120$ $2$ $2$ $7$
120.288.7-120.bdx.1.24 $120$ $2$ $2$ $7$
120.432.15-60.u.2.3 $120$ $3$ $3$ $15$