Properties

Label 8452.a.16904.1
Conductor 8452
Discriminant 16904
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

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, -3, -2, 1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, -3, -2, 1]), R([1, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 6, 5, -10, -6, 4, 1]))
 

Invariants

Conductor: \( N \)  =  \(8452\) = \( 2^{2} \cdot 2113 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(16904\) = \( 2^{3} \cdot 2113 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1800\) =  \( 2^{3} \cdot 3^{2} \cdot 5^{2} \)
\( I_4 \)  = \(80772\) =  \( 2^{2} \cdot 3 \cdot 53 \cdot 127 \)
\( I_6 \)  = \(43043976\) =  \( 2^{3} \cdot 3^{2} \cdot 597833 \)
\( I_{10} \)  = \(69238784\) =  \( 2^{15} \cdot 2113 \)
\( J_2 \)  = \(225\) =  \( 3^{2} \cdot 5^{2} \)
\( J_4 \)  = \(1268\) =  \( 2^{2} \cdot 317 \)
\( J_6 \)  = \(4224\) =  \( 2^{7} \cdot 3 \cdot 11 \)
\( J_8 \)  = \(-164356\) =  \( - 2^{2} \cdot 17 \cdot 2417 \)
\( J_{10} \)  = \(16904\) =  \( 2^{3} \cdot 2113 \)
\( g_1 \)  = \(576650390625/16904\)
\( g_2 \)  = \(3610828125/4226\)
\( g_3 \)  = \(26730000/2113\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : -1 : 1)\) \((-1 : -1 : 2)\) \((1 : -2 : 1)\) \((-1 : -2 : 2)\) \((2 : -2 : 1)\) \((2 : -9 : 1)\)
\((-4 : -2261 : 21)\) \((-4 : -5172 : 21)\)

magma: [C![-4,-5172,21],C![-4,-2261,21],C![-1,-2,2],C![-1,-1,2],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-9,1],C![2,-2,1]];
 

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
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.105278\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.060485\) \(\infty\)

2-torsion field: 6.6.1081856.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.006277 \)
Real period: \( 23.65540 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.445471 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(3\) \(2\) \(3\) \(1 + T + T^{2}\)
\(2113\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 34 T + 2113 T^{2} )\)

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 \(\Q\).