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Properties

Label 140.60.2-20.c.1.2
Level $140$
Index $60$
Genus $2$
Cusps $3$
$\Q$-cusps $3$

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Invariants

Level: $140$ $\SL_2$-level: $20$ Newform level: $50$
Index: $60$ $\PSL_2$-index:$30$
Genus: $2 = 1 + \frac{ 30 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$
Cusps: $3$ (all of which are rational) Cusp widths $5^{2}\cdot20$ Cusp orbits $1^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $3$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20A2

Level structure

$\GL_2(\Z/140\Z)$-generators: $\begin{bmatrix}66&73\\89&86\end{bmatrix}$, $\begin{bmatrix}77&102\\120&123\end{bmatrix}$, $\begin{bmatrix}78&83\\19&122\end{bmatrix}$, $\begin{bmatrix}117&6\\68&43\end{bmatrix}$
Contains $-I$: no $\quad$ (see 20.30.2.c.1 for the level structure with $-I$)
Cyclic 140-isogeny field degree: $48$
Cyclic 140-torsion field degree: $2304$
Full 140-torsion field degree: $1548288$

Models

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

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

$ 0 $ $=$ $ 9 x^{5} - 6 x^{4} y + x^{3} y^{2} + 10 x^{3} z^{2} - 3 x^{2} y z^{2} - x y^{2} z^{2} + 5 x z^{4} + y z^{4} $
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Weierstrass model Weierstrass model

$ y^{2} + \left(x^{2} + x\right) y $ $=$ $ x^{6} - 3x^{5} + 7x^{4} - 10x^{3} + 7x^{2} - 3x + 1 $
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Rational points

This modular curve has 3 rational cusps but no known non-cuspidal rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve CM $j$-invariant $j$-heightPlane modelWeierstrass modelEmbedded model
no$\infty$ $0.000$
32.a3 $-4$$1728$ $= 2^{6} \cdot 3^{3}$$7.455$$(0:0:1)$$(1:-1:0)$$(0:-2:1:0)$
32.a1 $-16$$287496$ $= 2^{3} \cdot 3^{3} \cdot 11^{3}$$12.569$$(1:3:1)$, $(-1:-3:1)$$(0:-1:1)$, $(1:0:1)$$(1/3:1/3:-1/6:1)$, $(1/3:-1/3:1/6:1)$

Maps to other modular curves

$j$-invariant map of degree 30 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{(3z^{2}-w^{2})^{3}}{z^{4}(2z-w)(2z+w)}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.30.2.c.1 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle y$

Equation of the image curve:

$0$ $=$ $ 9X^{5}-6X^{4}Y+X^{3}Y^{2}+10X^{3}Z^{2}-3X^{2}YZ^{2}-XY^{2}Z^{2}+5XZ^{4}+YZ^{4} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 20.30.2.c.1 :

$\displaystyle X$ $=$ $\displaystyle \frac{1}{2}xy-\frac{1}{2}y^{2}$
$\displaystyle Y$ $=$ $\displaystyle -\frac{3}{4}x^{4}y^{2}-\frac{3}{8}x^{3}y^{3}+\frac{1}{4}x^{3}y^{2}w+\frac{1}{8}x^{2}y^{4}-\frac{1}{8}xy^{5}-\frac{1}{4}xy^{4}w+\frac{1}{8}y^{6}$
$\displaystyle Z$ $=$ $\displaystyle xy$

Modular covers

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

Factor curve Level Index Degree Genus Rank
$X_{S_4}(5)$ $5$ $12$ $6$ $0$ $0$
28.12.0-4.c.1.2 $28$ $5$ $5$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
28.12.0-4.c.1.2 $28$ $5$ $5$ $0$ $0$

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

Covering curve Level Index Degree Genus
140.120.4-20.b.1.1 $140$ $2$ $2$ $4$
140.120.4-20.h.1.1 $140$ $2$ $2$ $4$
140.120.4-20.k.1.1 $140$ $2$ $2$ $4$
140.120.4-140.k.1.1 $140$ $2$ $2$ $4$
140.120.4-20.l.1.1 $140$ $2$ $2$ $4$
140.120.4-140.l.1.2 $140$ $2$ $2$ $4$
140.120.4-140.o.1.2 $140$ $2$ $2$ $4$
140.120.4-140.p.1.2 $140$ $2$ $2$ $4$
140.180.4-20.c.1.4 $140$ $3$ $3$ $4$
140.240.5-20.m.1.2 $140$ $4$ $4$ $5$
140.480.18-140.c.1.9 $140$ $8$ $8$ $18$
280.120.4-40.f.1.2 $280$ $2$ $2$ $4$
280.120.4-40.w.1.2 $280$ $2$ $2$ $4$
280.120.4-40.be.1.2 $280$ $2$ $2$ $4$
280.120.4-280.be.1.4 $280$ $2$ $2$ $4$
280.120.4-40.bh.1.2 $280$ $2$ $2$ $4$
280.120.4-280.bh.1.4 $280$ $2$ $2$ $4$
280.120.4-40.bk.1.8 $280$ $2$ $2$ $4$
280.120.4-40.bk.1.9 $280$ $2$ $2$ $4$
280.120.4-40.bl.1.15 $280$ $2$ $2$ $4$
280.120.4-40.bl.1.18 $280$ $2$ $2$ $4$
280.120.4-40.bm.1.7 $280$ $2$ $2$ $4$
280.120.4-40.bm.1.10 $280$ $2$ $2$ $4$
280.120.4-40.bn.1.5 $280$ $2$ $2$ $4$
280.120.4-40.bn.1.12 $280$ $2$ $2$ $4$
280.120.4-40.bo.1.5 $280$ $2$ $2$ $4$
280.120.4-40.bo.1.12 $280$ $2$ $2$ $4$
280.120.4-40.bp.1.7 $280$ $2$ $2$ $4$
280.120.4-40.bp.1.10 $280$ $2$ $2$ $4$
280.120.4-40.bq.1.7 $280$ $2$ $2$ $4$
280.120.4-40.bq.1.10 $280$ $2$ $2$ $4$
280.120.4-280.bq.1.4 $280$ $2$ $2$ $4$
280.120.4-40.br.1.8 $280$ $2$ $2$ $4$
280.120.4-40.br.1.9 $280$ $2$ $2$ $4$
280.120.4-280.bt.1.4 $280$ $2$ $2$ $4$
280.120.4-280.bw.1.16 $280$ $2$ $2$ $4$
280.120.4-280.bw.1.17 $280$ $2$ $2$ $4$
280.120.4-280.bx.1.12 $280$ $2$ $2$ $4$
280.120.4-280.bx.1.21 $280$ $2$ $2$ $4$
280.120.4-280.by.1.8 $280$ $2$ $2$ $4$
280.120.4-280.by.1.25 $280$ $2$ $2$ $4$
280.120.4-280.bz.1.4 $280$ $2$ $2$ $4$
280.120.4-280.bz.1.29 $280$ $2$ $2$ $4$
280.120.4-280.ca.1.4 $280$ $2$ $2$ $4$
280.120.4-280.ca.1.29 $280$ $2$ $2$ $4$
280.120.4-280.cb.1.8 $280$ $2$ $2$ $4$
280.120.4-280.cb.1.25 $280$ $2$ $2$ $4$
280.120.4-280.cc.1.12 $280$ $2$ $2$ $4$
280.120.4-280.cc.1.21 $280$ $2$ $2$ $4$
280.120.4-280.cd.1.16 $280$ $2$ $2$ $4$
280.120.4-280.cd.1.17 $280$ $2$ $2$ $4$