Properties

Label 6982.a.13964.1
Conductor 6982
Discriminant -13964
Mordell-Weil group \(\Z \times \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = x^5 - 3x^3 + 2x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - 3x^3z^3 + 2xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 10x^3 + 8x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 0, -3, 0, 1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 0, -3, 0, 1]), R([1, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 8, 0, -10, 0, 4, 1]))
 

Invariants

Conductor: \( N \)  =  \(6982\) = \( 2 \cdot 3491 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-13964\) = \( - 2^{2} \cdot 3491 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1640\) =  \( 2^{3} \cdot 5 \cdot 41 \)
\( I_4 \)  = \(50020\) =  \( 2^{2} \cdot 5 \cdot 41 \cdot 61 \)
\( I_6 \)  = \(23554280\) =  \( 2^{3} \cdot 5 \cdot 263 \cdot 2239 \)
\( I_{10} \)  = \(-57196544\) =  \( - 2^{14} \cdot 3491 \)
\( J_2 \)  = \(205\) =  \( 5 \cdot 41 \)
\( J_4 \)  = \(1230\) =  \( 2 \cdot 3 \cdot 5 \cdot 41 \)
\( J_6 \)  = \(8720\) =  \( 2^{4} \cdot 5 \cdot 109 \)
\( J_8 \)  = \(68675\) =  \( 5^{2} \cdot 41 \cdot 67 \)
\( J_{10} \)  = \(-13964\) =  \( - 2^{2} \cdot 3491 \)
\( g_1 \)  = \(-362050628125/13964\)
\( g_2 \)  = \(-5298301875/6982\)
\( g_3 \)  = \(-91614500/3491\)

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

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((1 : -2 : 1)\) \((-2 : 3 : 1)\) \((-2 : 4 : 1)\) \((-3 : 12 : 1)\) \((-3 : 14 : 1)\) \((13 : 408 : 5)\)
\((13 : -2730 : 5)\)

magma: [C![-3,12,1],C![-3,14,1],C![-2,3,1],C![-2,4,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![13,-2730,5],C![13,408,5]];
 

Number of rational Weierstrass points: \(1\)

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.474383\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.015759\) \(\infty\)

2-torsion field: 5.3.55856.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.007467 \)
Real period: \( 23.52533 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.351346 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(3491\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 92 T + 3491 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

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