Properties

Label 6463.a.6463.1
Conductor 6463
Discriminant -6463
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

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

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

Invariants

Conductor: \( N \)  =  \(6463\) = \( 23 \cdot 281 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-6463\) = \( - 23 \cdot 281 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-792\) =  \( - 2^{3} \cdot 3^{2} \cdot 11 \)
\( I_4 \)  = \(29988\) =  \( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 17 \)
\( I_6 \)  = \(-7490520\) =  \( - 2^{3} \cdot 3^{2} \cdot 5 \cdot 20807 \)
\( I_{10} \)  = \(-26472448\) =  \( - 2^{12} \cdot 23 \cdot 281 \)
\( J_2 \)  = \(-99\) =  \( - 3^{2} \cdot 11 \)
\( J_4 \)  = \(96\) =  \( 2^{5} \cdot 3 \)
\( J_6 \)  = \(2168\) =  \( 2^{3} \cdot 271 \)
\( J_8 \)  = \(-55962\) =  \( - 2 \cdot 3^{2} \cdot 3109 \)
\( J_{10} \)  = \(-6463\) =  \( - 23 \cdot 281 \)
\( g_1 \)  = \(9509900499/6463\)
\( g_2 \)  = \(93148704/6463\)
\( g_3 \)  = \(-21248568/6463\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : -1 : 1)\) \((1 : -3 : 1)\) \((-2 : -5 : 3)\) \((-2 : -16 : 3)\) \((-6 : -112 : 13)\) \((-6 : -1539 : 13)\)

magma: [C![-6,-1539,13],C![-6,-112,13],C![-2,-16,3],C![-2,-5,3],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,1],C![1,1,0]];
 

Number of rational Weierstrass points: \(0\)

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
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.124959\) \(\infty\)
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.121215\) \(\infty\)

2-torsion field: 6.0.413632.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.014972 \)
Real period: \( 22.48847 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.336705 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(23\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T + 23 T^{2} )\)
\(281\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 10 T + 281 T^{2} )\)

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 \(\Q\).