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Properties

Label 10028.a.641792.1
Conductor $10028$
Discriminant $-641792$
Mordell-Weil group \(\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x)y = 2x^4 + 2x^3 + 6x^2 + 2x + 4$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2)y = 2x^4z^2 + 2x^3z^3 + 6x^2z^4 + 2xz^5 + 4z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 11x^4 + 10x^3 + 25x^2 + 8x + 16$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 2, 6, 2, 2]), R([0, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 2, 6, 2, 2], R![0, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([16, 8, 25, 10, 11, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10028\) \(=\) \( 2^{2} \cdot 23 \cdot 109 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-641792\) \(=\) \( - 2^{8} \cdot 23 \cdot 109 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(3500\) \(=\)  \( 2^{2} \cdot 5^{3} \cdot 7 \)
\( I_4 \)  \(=\) \(34537\) \(=\)  \( 34537 \)
\( I_6 \)  \(=\) \(37868843\) \(=\)  \( 17 \cdot 19 \cdot 117241 \)
\( I_{10} \)  \(=\) \(82149376\) \(=\)  \( 2^{15} \cdot 23 \cdot 109 \)
\( J_2 \)  \(=\) \(875\) \(=\)  \( 5^{3} \cdot 7 \)
\( J_4 \)  \(=\) \(30462\) \(=\)  \( 2 \cdot 3 \cdot 5077 \)
\( J_6 \)  \(=\) \(1374556\) \(=\)  \( 2^{2} \cdot 343639 \)
\( J_8 \)  \(=\) \(68700764\) \(=\)  \( 2^{2} \cdot 11 \cdot 919 \cdot 1699 \)
\( J_{10} \)  \(=\) \(641792\) \(=\)  \( 2^{8} \cdot 23 \cdot 109 \)
\( g_1 \)  \(=\) \(512908935546875/641792\)
\( g_2 \)  \(=\) \(10203580078125/320896\)
\( g_3 \)  \(=\) \(263098609375/160448\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : -2 : 1),\, (0 : 2 : 1),\, (-2 : -4 : 1),\, (-2 : 10 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : -2 : 1),\, (0 : 2 : 1),\, (-2 : -4 : 1),\, (-2 : 10 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -4 : 1),\, (0 : 4 : 1),\, (-2 : -14 : 1),\, (-2 : 14 : 1)\)

magma: [C![-2,-4,1],C![-2,10,1],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![-2,-14,1],C![-2,14,1],C![0,-4,1],C![0,4,1],C![1,-1,0],C![1,1,0]]; // simplified model
 

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -2 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0.007912\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -2 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0.007912\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -4 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + x^2z + xz^2 - 4z^3\) \(0.007912\) \(\infty\)

2-torsion field: 6.0.641792.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.007912 \)
Real period: \( 7.935388 \)
Tamagawa product: \( 13 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.816245 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(8\) \(13\) \(( 1 - T )( 1 + T )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 23 T^{2} )\)
\(109\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 7 T + 109 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

Sato-Tate group

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

Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);