Properties

Label 10080.c.141120.1
Conductor $10080$
Discriminant $141120$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{4}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -x^6 + 35x^4 - 560x^2 + 2940$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 + 35x^4z^2 - 560x^2z^4 + 2940z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 142x^4 - 2239x^2 + 11760$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2940, 0, -560, 0, 35, 0, -1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2940, 0, -560, 0, 35, 0, -1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([11760, 0, -2239, 0, 142, 0, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10080\) \(=\) \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(141120\) \(=\) \( 2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(3388552\) \(=\)  \( 2^{3} \cdot 467 \cdot 907 \)
\( I_4 \)  \(=\) \(174712\) \(=\)  \( 2^{3} \cdot 21839 \)
\( I_6 \)  \(=\) \(197326050612\) \(=\)  \( 2^{2} \cdot 3 \cdot 227 \cdot 2713 \cdot 26701 \)
\( I_{10} \)  \(=\) \(564480\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
\( J_2 \)  \(=\) \(1694276\) \(=\)  \( 2^{2} \cdot 467 \cdot 907 \)
\( J_4 \)  \(=\) \(119607102722\) \(=\)  \( 2 \cdot 23 \cdot 6211 \cdot 418637 \)
\( J_6 \)  \(=\) \(11258185829425920\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11 \cdot 1813122589 \)
\( J_8 \)  \(=\) \(1192153758196342556159\) \(=\)  \( 1619 \cdot 2861 \cdot 902767 \cdot 285096503 \)
\( J_{10} \)  \(=\) \(141120\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
\( g_1 \)  \(=\) \(218142768611210403574323981584/2205\)
\( g_2 \)  \(=\) \(9089279812657801356650662498/2205\)
\( g_3 \)  \(=\) \(229006686528379459553216\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((-4 : 34 : 1),\, (4 : -34 : 1)\)
All points: \((-4 : 34 : 1),\, (4 : -34 : 1)\)
All points: \((-4 : 0 : 1),\, (4 : 0 : 1)\)

magma: [C![-4,34,1],C![4,-34,1]]; // minimal model
 
magma: [C![-4,0,1],C![4,0,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \times \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-4 : 34 : 1) + (4 : -34 : 1) - D_\infty\) \((x - 4z) (x + 4z)\) \(=\) \(0,\) \(2y\) \(=\) \(-17xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(13x^2 - 210z^2\) \(=\) \(0,\) \(13y\) \(=\) \(-111xz^2\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-4 : 34 : 1) + (4 : -34 : 1) - D_\infty\) \((x - 4z) (x + 4z)\) \(=\) \(0,\) \(2y\) \(=\) \(-17xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(13x^2 - 210z^2\) \(=\) \(0,\) \(13y\) \(=\) \(-111xz^2\) \(0\) \(4\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \((x - 4z) (x + 4z)\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 - 33xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(13x^2 - 210z^2\) \(=\) \(0,\) \(13y\) \(=\) \(x^3 - 221xz^2\) \(0\) \(4\)

2-torsion field: \(\Q(\sqrt{3}, \sqrt{5})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(4\)
Regulator: \( 1 \)
Real period: \( 5.655296 \)
Tamagawa product: \( 4 \)
Torsion order:\( 8 \)
Leading coefficient: \( 1.413824 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(6\) \(2\) \(1 - T\)
\(3\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(7\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + 7 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 210.c
  Elliptic curve isogeny class 48.a

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).