Properties

Label 8204.a.32816.1
Conductor 8204
Discriminant 32816
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

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

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

Invariants

Conductor: \( N \)  =  \(8204\) = \( 2^{2} \cdot 7 \cdot 293 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(32816\) = \( 2^{4} \cdot 7 \cdot 293 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-576\) =  \( - 2^{6} \cdot 3^{2} \)
\( I_4 \)  = \(22848\) =  \( 2^{6} \cdot 3 \cdot 7 \cdot 17 \)
\( I_6 \)  = \(-4981248\) =  \( - 2^{9} \cdot 3^{2} \cdot 23 \cdot 47 \)
\( I_{10} \)  = \(134414336\) =  \( 2^{16} \cdot 7 \cdot 293 \)
\( J_2 \)  = \(-72\) =  \( - 2^{3} \cdot 3^{2} \)
\( J_4 \)  = \(-22\) =  \( - 2 \cdot 11 \)
\( J_6 \)  = \(3024\) =  \( 2^{4} \cdot 3^{3} \cdot 7 \)
\( J_8 \)  = \(-54553\) =  \( - 17 \cdot 3209 \)
\( J_{10} \)  = \(32816\) =  \( 2^{4} \cdot 7 \cdot 293 \)
\( g_1 \)  = \(-120932352/2051\)
\( g_2 \)  = \(513216/2051\)
\( g_3 \)  = \(139968/293\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\)
\((-1 : 2 : 1)\) \((-1 : -3 : 1)\) \((1 : -3 : 2)\) \((1 : -5 : 2)\) \((-2 : 9 : 1)\) \((-2 : -10 : 1)\)
\((3 : -223 : 10)\) \((3 : -777 : 10)\)

magma: [C![-2,-10,1],C![-2,9,1],C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-3,2],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![3,-777,10],C![3,-223,10]];
 

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
\((0 : -1 : 1) + (1 : -5 : 2) - (1 : -1 : 0) - (1 : 1 : 0)\) \((-2x + z) x\) \(=\) \(0,\) \(4y\) \(=\) \(3xz^2 - 4z^3\) \(0.169537\) \(\infty\)
\((1 : -1 : 1) - (1 : 1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.040146\) \(\infty\)

2-torsion field: 6.2.525056.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.006769 \)
Real period: \( 18.14031 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.368388 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(2\) \(3\) \(1 + 2 T + 2 T^{2}\)
\(7\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 7 T^{2} )\)
\(293\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 19 T + 293 T^{2} )\)

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 \(\Q\).