↑

Properties

Label 120.96.2-60.a.1.6
Level $120$
Index $96$
Genus $2$
Cusps $6$
$\Q$-cusps $6$

Related objects

Downloads

Learn more

Invariants

Level: $120$ $\SL_2$-level: $12$ Newform level: $1200$
Index: $96$ $\PSL_2$-index:$48$
Genus: $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $4^{3}\cdot12^{3}$ Cusp orbits $1^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12F2

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}59&52\\66&55\end{bmatrix}$, $\begin{bmatrix}73&116\\12&41\end{bmatrix}$, $\begin{bmatrix}89&6\\84&95\end{bmatrix}$, $\begin{bmatrix}93&14\\82&29\end{bmatrix}$, $\begin{bmatrix}97&30\\22&101\end{bmatrix}$, $\begin{bmatrix}119&106\\72&25\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.48.2.a.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree: $24$
Cyclic 120-torsion field degree: $768$
Full 120-torsion field degree: $368640$

Models

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

$ 0 $ $=$ $ 2 x y w - x z w + 4 y z w + z^{2} w $
$=$ $2 x y^{2} - x y z + 4 y^{2} z + y z^{2}$
$=$ $2 x^{2} y - x^{2} z + 4 x y z + x z^{2}$
$=$ $2 x y z - x z^{2} + 4 y z^{2} + z^{3}$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 5 x^{3} z + 2 x^{2} y^{2} + 15 x^{2} z^{2} - x y^{2} z + 10 x z^{3} - y^{2} z^{2} $
 Copy content Toggle raw display

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ 10x^{5} + 25x^{4} - 25x^{2} - 10x $
 Copy content Toggle raw display

Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{10628820000x^{10}-11337408000x^{8}w^{2}+5290790400x^{6}w^{4}-1662819840x^{4}w^{6}+366156288x^{2}w^{8}-18858774375xz^{9}+17994636000xz^{7}w^{2}+1822726800xz^{5}w^{4}+2482839360xz^{3}w^{6}-451816704xzw^{8}+8398080000y^{10}-7464960000y^{8}w^{2}+5034700800y^{6}w^{4}-1789378560y^{4}w^{6}+541657088y^{2}w^{8}+35618028750yz^{9}-54971703000yz^{7}w^{2}-16851499200yz^{5}w^{4}-7847049600yz^{3}w^{6}+809435136yzw^{8}+8361174375z^{10}-19024348500z^{8}w^{2}-4871988000z^{6}w^{4}-2424795840z^{4}w^{6}+169345024z^{2}w^{8}}{w^{4}(3954825xz^{5}-111420xz^{3}w^{2}-4147200y^{6}+2211840y^{4}w^{2}-184320y^{2}w^{4}-12121650yz^{5}-937440yz^{3}w^{2}-4096yzw^{4}-4019625z^{6}-181080z^{4}w^{2}+10496z^{2}w^{4})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.48.2.a.1 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}z$

Equation of the image curve:

$0$ $=$ $ 2X^{2}Y^{2}+5X^{3}Z-XY^{2}Z+15X^{2}Z^{2}-Y^{2}Z^{2}+10XZ^{3} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 60.48.2.a.1 :

$\displaystyle X$ $=$ $\displaystyle \frac{1}{2}z$
$\displaystyle Y$ $=$ $\displaystyle -\frac{2}{3}y^{2}w+\frac{1}{6}yzw+\frac{1}{12}z^{2}w$
$\displaystyle Z$ $=$ $\displaystyle y$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
24.48.0-6.a.1.5 $24$ $2$ $2$ $0$ $0$
120.48.0-6.a.1.7 $120$ $2$ $2$ $0$ $?$

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

Covering curve Level Index Degree Genus
120.192.3-60.k.1.1 $120$ $2$ $2$ $3$
120.192.3-60.k.1.16 $120$ $2$ $2$ $3$
120.192.3-60.l.1.12 $120$ $2$ $2$ $3$
120.192.3-60.l.1.30 $120$ $2$ $2$ $3$
120.192.3-60.l.1.31 $120$ $2$ $2$ $3$
120.192.3-60.m.1.8 $120$ $2$ $2$ $3$
120.192.3-60.m.1.9 $120$ $2$ $2$ $3$
120.192.3-60.m.1.16 $120$ $2$ $2$ $3$
120.192.3-60.n.1.8 $120$ $2$ $2$ $3$
120.192.3-60.n.1.11 $120$ $2$ $2$ $3$
120.192.3-60.n.1.14 $120$ $2$ $2$ $3$
120.192.3-60.o.1.8 $120$ $2$ $2$ $3$
120.192.3-60.o.1.11 $120$ $2$ $2$ $3$
120.192.3-60.o.1.14 $120$ $2$ $2$ $3$
120.192.3-60.p.1.8 $120$ $2$ $2$ $3$
120.192.3-60.p.1.9 $120$ $2$ $2$ $3$
120.192.3-60.p.1.16 $120$ $2$ $2$ $3$
120.192.3-60.q.1.5 $120$ $2$ $2$ $3$
120.192.3-60.q.1.12 $120$ $2$ $2$ $3$
120.192.3-60.q.1.16 $120$ $2$ $2$ $3$
120.192.3-60.r.1.4 $120$ $2$ $2$ $3$
120.192.3-60.r.1.13 $120$ $2$ $2$ $3$
120.192.3-60.r.1.16 $120$ $2$ $2$ $3$
120.192.3-120.fa.1.5 $120$ $2$ $2$ $3$
120.192.3-120.fa.1.21 $120$ $2$ $2$ $3$
120.192.3-120.fa.1.28 $120$ $2$ $2$ $3$
120.192.3-120.fd.1.10 $120$ $2$ $2$ $3$
120.192.3-120.fd.1.20 $120$ $2$ $2$ $3$
120.192.3-120.fd.1.21 $120$ $2$ $2$ $3$
120.192.3-120.fg.1.1 $120$ $2$ $2$ $3$
120.192.3-120.fg.1.25 $120$ $2$ $2$ $3$
120.192.3-120.fg.1.32 $120$ $2$ $2$ $3$
120.192.3-120.fj.1.8 $120$ $2$ $2$ $3$
120.192.3-120.fj.1.9 $120$ $2$ $2$ $3$
120.192.3-120.fj.1.20 $120$ $2$ $2$ $3$
120.192.3-120.fm.1.1 $120$ $2$ $2$ $3$
120.192.3-120.fm.1.27 $120$ $2$ $2$ $3$
120.192.3-120.fm.1.30 $120$ $2$ $2$ $3$
120.192.3-120.fp.1.7 $120$ $2$ $2$ $3$
120.192.3-120.fp.1.10 $120$ $2$ $2$ $3$
120.192.3-120.fp.1.23 $120$ $2$ $2$ $3$
120.192.3-120.fs.1.3 $120$ $2$ $2$ $3$
120.192.3-120.fs.1.23 $120$ $2$ $2$ $3$
120.192.3-120.fs.1.26 $120$ $2$ $2$ $3$
120.192.3-120.fv.1.11 $120$ $2$ $2$ $3$
120.192.3-120.fv.1.19 $120$ $2$ $2$ $3$
120.192.3-120.fv.1.22 $120$ $2$ $2$ $3$
120.288.7-60.o.1.6 $120$ $3$ $3$ $7$
120.480.18-60.a.2.2 $120$ $5$ $5$ $18$