Properties

Label 539344.a.539344.1
Conductor $539344$
Discriminant $-539344$
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

$y^2 + xy = -x^6 - 2x^5 - 11x^4 - 16x^3 - 40x^2 - 32x - 47$ (homogenize, simplify)
$y^2 + xz^2y = -x^6 - 2x^5z - 11x^4z^2 - 16x^3z^3 - 40x^2z^4 - 32xz^5 - 47z^6$ (dehomogenize, simplify)
$y^2 = -4x^6 - 8x^5 - 44x^4 - 64x^3 - 159x^2 - 128x - 188$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-47, -32, -40, -16, -11, -2, -1]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-47, -32, -40, -16, -11, -2, -1], R![0, 1]);
 
sage: X = HyperellipticCurve(R([-188, -128, -159, -64, -44, -8, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(539344\) \(=\) \( 2^{4} \cdot 13 \cdot 2593 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-539344\) \(=\) \( - 2^{4} \cdot 13 \cdot 2593 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(56720\) \(=\)  \( 2^{4} \cdot 5 \cdot 709 \)
\( I_4 \)  \(=\) \(4496440\) \(=\)  \( 2^{3} \cdot 5 \cdot 13 \cdot 8647 \)
\( I_6 \)  \(=\) \(81835622676\) \(=\)  \( 2^{2} \cdot 3^{3} \cdot 31 \cdot 24443137 \)
\( I_{10} \)  \(=\) \(2157376\) \(=\)  \( 2^{6} \cdot 13 \cdot 2593 \)
\( J_2 \)  \(=\) \(28360\) \(=\)  \( 2^{3} \cdot 5 \cdot 709 \)
\( J_4 \)  \(=\) \(32762660\) \(=\)  \( 2^{2} \cdot 5 \cdot 7 \cdot 31 \cdot 7549 \)
\( J_6 \)  \(=\) \(49610935036\) \(=\)  \( 2^{2} \cdot 13 \cdot 19 \cdot 50213497 \)
\( J_8 \)  \(=\) \(83393556836340\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 71 \cdot 547 \cdot 35787847 \)
\( J_{10} \)  \(=\) \(539344\) \(=\)  \( 2^{4} \cdot 13 \cdot 2593 \)
\( g_1 \)  \(=\) \(1146597920784313600000/33709\)
\( g_2 \)  \(=\) \(46706556736980560000/33709\)
\( g_3 \)  \(=\) \(191834418729473200/2593\)

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 $\R$.

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

2-torsion field: 6.4.4545186724.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(4\)
Regulator: \( 1 \)
Real period: \( 2.567917 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 5.135835 \)
Analytic order of Ш: \( 8 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(4\) \(1\) \(1 - T\)
\(13\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 13 T^{2} )\)
\(2593\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 18 T + 2593 T^{2} )\)

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);