↑

Properties

Label 120.72.2-24.c.1.1
Level $120$
Index $72$
Genus $2$
Cusps $4$
$\Q$-cusps $2$

Related objects

Downloads

Learn more

Invariants

Level: $120$ $\SL_2$-level: $12$ Newform level: $576$
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: $\begin{bmatrix}29&112\\106&27\end{bmatrix}$, $\begin{bmatrix}49&0\\4&77\end{bmatrix}$, $\begin{bmatrix}73&40\\102&47\end{bmatrix}$, $\begin{bmatrix}77&52\\52&63\end{bmatrix}$, $\begin{bmatrix}115&114\\18&43\end{bmatrix}$, $\begin{bmatrix}119&56\\84&67\end{bmatrix}$
Contains $-I$: no $\quad$ (see 24.36.2.c.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree: $96$
Cyclic 120-torsion field degree: $3072$
Full 120-torsion field degree: $491520$

Models

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

$ 0 $ $=$ $ 6 x^{2} y + x z^{2} + y w^{2} $
$=$ $6 x^{2} w + 24 x y z + w^{3}$
$=$ $24 y^{2} w - z w^{2}$
$=$ $24 y^{2} z - z^{2} w$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ x^{3} y + 27 y^{2} z^{2} + 2 z^{4} $
 Copy content Toggle raw display

Weierstrass model Weierstrass model

$ y^{2} + x^{3} y $ $=$ $ -54 $
 Copy content Toggle raw display

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{216x^{7}w+72x^{5}w^{3}+18x^{3}w^{5}+xw^{7}-64yz^{7}+32yz^{4}w^{3}-8yzw^{6}}{w^{5}(6x^{3}+xw^{2}-4yzw)}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.36.2.c.1 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{18}x$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{12}w$

Equation of the image curve:

$0$ $=$ $ X^{3}Y+27Y^{2}Z^{2}+2Z^{4} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.36.2.c.1 :

$\displaystyle X$ $=$ $\displaystyle -y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{96}xw^{2}+y^{3}$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{12}w$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
60.36.1-6.a.1.6 $60$ $2$ $2$ $1$ $0$
120.24.0-24.a.1.8 $120$ $3$ $3$ $0$ $?$
120.36.1-6.a.1.2 $120$ $2$ $2$ $1$ $?$

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

Covering curve Level Index Degree Genus
120.144.3-24.m.1.2 $120$ $2$ $2$ $3$
120.144.3-24.o.1.2 $120$ $2$ $2$ $3$
120.144.3-24.s.1.4 $120$ $2$ $2$ $3$
120.144.3-24.u.1.4 $120$ $2$ $2$ $3$
120.144.3-24.bl.1.2 $120$ $2$ $2$ $3$
120.144.3-24.bm.1.2 $120$ $2$ $2$ $3$
120.144.3-24.bo.1.2 $120$ $2$ $2$ $3$
120.144.3-24.bp.1.2 $120$ $2$ $2$ $3$
120.144.3-120.bp.1.4 $120$ $2$ $2$ $3$
120.144.3-120.bq.1.8 $120$ $2$ $2$ $3$
120.144.3-120.bs.1.13 $120$ $2$ $2$ $3$
120.144.3-120.bt.1.12 $120$ $2$ $2$ $3$
120.144.3-120.cz.1.11 $120$ $2$ $2$ $3$
120.144.3-120.da.1.9 $120$ $2$ $2$ $3$
120.144.3-120.dc.1.6 $120$ $2$ $2$ $3$
120.144.3-120.dd.1.5 $120$ $2$ $2$ $3$
120.144.4-24.d.1.10 $120$ $2$ $2$ $4$
120.144.4-24.e.1.2 $120$ $2$ $2$ $4$
120.144.4-24.e.1.12 $120$ $2$ $2$ $4$
120.144.4-24.h.1.2 $120$ $2$ $2$ $4$
120.144.4-24.h.1.22 $120$ $2$ $2$ $4$
120.144.4-24.j.1.2 $120$ $2$ $2$ $4$
120.144.4-24.j.1.10 $120$ $2$ $2$ $4$
120.144.4-120.p.1.10 $120$ $2$ $2$ $4$
120.144.4-120.p.1.22 $120$ $2$ $2$ $4$
120.144.4-120.q.1.12 $120$ $2$ $2$ $4$
120.144.4-120.q.1.18 $120$ $2$ $2$ $4$
120.144.4-120.s.1.8 $120$ $2$ $2$ $4$
120.144.4-120.s.1.18 $120$ $2$ $2$ $4$
120.144.4-120.t.1.6 $120$ $2$ $2$ $4$
120.144.4-120.t.1.18 $120$ $2$ $2$ $4$
120.144.4-24.bp.1.4 $120$ $2$ $2$ $4$
120.144.4-24.bp.1.8 $120$ $2$ $2$ $4$
120.144.4-24.bq.1.2 $120$ $2$ $2$ $4$
120.144.4-24.bq.1.8 $120$ $2$ $2$ $4$
120.144.4-24.bs.1.6 $120$ $2$ $2$ $4$
120.144.4-24.bs.1.16 $120$ $2$ $2$ $4$
120.144.4-24.bt.1.12 $120$ $2$ $2$ $4$
120.144.4-24.bt.1.14 $120$ $2$ $2$ $4$
120.144.4-120.cf.1.10 $120$ $2$ $2$ $4$
120.144.4-120.cf.1.18 $120$ $2$ $2$ $4$
120.144.4-120.cg.1.14 $120$ $2$ $2$ $4$
120.144.4-120.cg.1.18 $120$ $2$ $2$ $4$
120.144.4-120.ci.1.14 $120$ $2$ $2$ $4$
120.144.4-120.ci.1.18 $120$ $2$ $2$ $4$
120.144.4-120.cj.1.10 $120$ $2$ $2$ $4$
120.144.4-120.cj.1.22 $120$ $2$ $2$ $4$
120.360.14-120.c.1.5 $120$ $5$ $5$ $14$
120.432.15-120.c.1.39 $120$ $6$ $6$ $15$