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Properties

Label 70.40.1.c.1
Level $70$
Index $40$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $0$

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Invariants

Level: $70$ $\SL_2$-level: $10$ Newform level: $20$
Index: $40$ $\PSL_2$-index:$40$
Genus: $1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $10^{4}$ Cusp orbits $4$
Elliptic points: $0$ of order $2$ and $4$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 70.40.1.11

Level structure

$\GL_2(\Z/70\Z)$-generators: $\begin{bmatrix}35&6\\47&65\end{bmatrix}$, $\begin{bmatrix}35&57\\66&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 70-isogeny field degree: $144$
Cyclic 70-torsion field degree: $3456$
Full 70-torsion field degree: $145152$

Jacobian

Conductor: $2^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 20.2.a.a

Models

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

$ 0 $ $=$ $ 8 y^{2} - y z - 2 y w - 2 z^{2} + 2 z w - 3 w^{2} $
$=$ $35 x^{2} + 3 y^{2} - y z - y w - z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 15 x^{3} z + 245 x^{2} y^{2} + 10 x^{2} z^{2} - 10 x z^{3} - 4900 y^{4} - 5 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

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

Maps to other modular curves

$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 5^2\,\frac{78170505375yz^{9}-156662816850yz^{8}w+249314710320yz^{7}w^{2}-204177569568yz^{6}w^{3}+146085522912yz^{5}w^{4}-51836710464yz^{4}w^{5}+20489704704yz^{3}w^{6}-420208128yz^{2}w^{7}+1049580288yzw^{8}+53761536yw^{9}+37170022750z^{10}-104566501150z^{9}w+200982069405z^{8}w^{2}-223757617712z^{7}w^{3}+192381620984z^{6}w^{4}-104847029280z^{5}w^{5}+47824205184z^{4}w^{6}-11364917760z^{3}w^{7}+3445849728z^{2}w^{8}+111449088zw^{9}+44575488w^{10}}{10705525yz^{9}+77998610yz^{8}w+233385360yz^{7}w^{2}+367786720yz^{6}w^{3}+333310880yz^{5}w^{4}+62036800yz^{4}w^{5}+6424320yz^{3}w^{6}+3473920yz^{2}w^{7}-1007360yzw^{8}+104960yw^{9}+4725322z^{10}+30600230z^{9}w+78682255z^{8}w^{2}+107171480z^{7}w^{3}+95027800z^{6}w^{4}+137603648z^{5}w^{5}+42020160z^{4}w^{6}+4735360z^{3}w^{7}-2285440z^{2}w^{8}+780800zw^{9}-78592w^{10}}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
10.20.1.b.1 $10$ $2$ $2$ $1$ $0$ dimension zero
35.20.0.a.1 $35$ $2$ $2$ $0$ $0$ full Jacobian
70.20.0.a.1 $70$ $2$ $2$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
70.120.5.n.1 $70$ $3$ $3$ $5$ $2$ $1^{4}$
70.120.5.r.1 $70$ $3$ $3$ $5$ $1$ $1^{4}$
70.320.21.f.1 $70$ $8$ $8$ $21$ $8$ $1^{14}\cdot2^{3}$
70.840.65.v.1 $70$ $21$ $21$ $65$ $34$ $1^{6}\cdot2^{9}\cdot3^{4}\cdot4^{7}$
70.1120.85.p.1 $70$ $28$ $28$ $85$ $42$ $1^{20}\cdot2^{12}\cdot3^{4}\cdot4^{7}$
140.160.9.i.1 $140$ $4$ $4$ $9$ $?$ not computed
210.120.9.bk.1 $210$ $3$ $3$ $9$ $?$ not computed
210.160.9.i.1 $210$ $4$ $4$ $9$ $?$ not computed