Properties

Label 6880.a.110080.1
Conductor $6880$
Discriminant $-110080$
Mordell-Weil group \(\Z/{2}\Z\)
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

$y^2 + y = -8x^6 - 16x^5 + 3x^4 + 19x^3 + 7x^2 - x - 1$ (homogenize, simplify)
$y^2 + z^3y = -8x^6 - 16x^5z + 3x^4z^2 + 19x^3z^3 + 7x^2z^4 - xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = -32x^6 - 64x^5 + 12x^4 + 76x^3 + 28x^2 - 4x - 3$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -1, 7, 19, 3, -16, -8]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -1, 7, 19, 3, -16, -8], R![1]);
 
sage: X = HyperellipticCurve(R([-3, -4, 28, 76, 12, -64, -32]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(6880\) \(=\) \( 2^{5} \cdot 5 \cdot 43 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-110080\) \(=\) \( - 2^{9} \cdot 5 \cdot 43 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1030\) \(=\)  \( 2 \cdot 5 \cdot 103 \)
\( I_4 \)  \(=\) \(158404\) \(=\)  \( 2^{2} \cdot 199^{2} \)
\( I_6 \)  \(=\) \(26682485\) \(=\)  \( 5 \cdot 5336497 \)
\( I_{10} \)  \(=\) \(-430\) \(=\)  \( - 2 \cdot 5 \cdot 43 \)
\( J_2 \)  \(=\) \(2060\) \(=\)  \( 2^{2} \cdot 5 \cdot 103 \)
\( J_4 \)  \(=\) \(-245594\) \(=\)  \( - 2 \cdot 19 \cdot 23 \cdot 281 \)
\( J_6 \)  \(=\) \(72206340\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 1019 \cdot 1181 \)
\( J_8 \)  \(=\) \(22107161891\) \(=\)  \( 101 \cdot 109 \cdot 131 \cdot 15329 \)
\( J_{10} \)  \(=\) \(-110080\) \(=\)  \( - 2^{9} \cdot 5 \cdot 43 \)
\( g_1 \)  \(=\) \(-14490925928750/43\)
\( g_2 \)  \(=\) \(3354589935475/172\)
\( g_3 \)  \(=\) \(-957546326325/344\)

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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model
 
magma: []; // simplified model
 

Number of rational Weierstrass points: \(0\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable except over $\Q_{2}$.

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: \(\Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(8x^2 + 8xz - 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(8x^2 + 8xz - 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(8x^2 + 8xz - 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)

2-torsion field: 6.2.473344000.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 1.370726 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 1.370726 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(9\) \(1\) \(1 - 2 T + 2 T^{2}\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 5 T^{2} )\)
\(43\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 43 T^{2} )\)

Galois representations

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

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.15.1 yes

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);