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Properties

Label 276083.a.276083.2
Conductor $276083$
Discriminant $-276083$
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

$y^2 + (x^3 + x)y = -x^6 - x^5 + 4x^4 + 8x^3 - 16x - 25$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 - x^5z + 4x^4z^2 + 8x^3z^3 - 16xz^5 - 25z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 - 4x^5 + 18x^4 + 32x^3 + x^2 - 64x - 100$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-25, -16, 0, 8, 4, -1, -1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-25, -16, 0, 8, 4, -1, -1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-100, -64, 1, 32, 18, -4, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(276083\) \(=\) \( 276083 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-276083\) \(=\) \( -276083 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(13976\) \(=\)  \( 2^{3} \cdot 1747 \)
\( I_4 \)  \(=\) \(4744408\) \(=\)  \( 2^{3} \cdot 593051 \)
\( I_6 \)  \(=\) \(25551167045\) \(=\)  \( 5 \cdot 31 \cdot 2347 \cdot 70237 \)
\( I_{10} \)  \(=\) \(1104332\) \(=\)  \( 2^{2} \cdot 276083 \)
\( J_2 \)  \(=\) \(6988\) \(=\)  \( 2^{2} \cdot 1747 \)
\( J_4 \)  \(=\) \(1243938\) \(=\)  \( 2 \cdot 3 \cdot 151 \cdot 1373 \)
\( J_6 \)  \(=\) \(-514209569\) \(=\)  \( - 9293 \cdot 55333 \)
\( J_8 \)  \(=\) \(-1285169554004\) \(=\)  \( - 2^{2} \cdot 7 \cdot 1163 \cdot 39465961 \)
\( J_{10} \)  \(=\) \(276083\) \(=\)  \( 276083 \)
\( g_1 \)  \(=\) \(16663433074005511168/276083\)
\( g_2 \)  \(=\) \(424480186886987136/276083\)
\( g_3 \)  \(=\) \(-25109955719585936/276083\)

  (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: \((-2 : 5 : 1),\, (2 : -5 : 1)\)
All points: \((-2 : 5 : 1),\, (2 : -5 : 1)\)
All points: \((-2 : 0 : 1),\, (2 : 0 : 1)\)

magma: [C![-2,5,1],C![2,-5,1]]; // minimal model
 
magma: [C![-2,0,1],C![2,0,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : 5 : 1) + (2 : -5 : 1) - D_\infty\) \((x - 2z) (x + 2z)\) \(=\) \(0,\) \(2y\) \(=\) \(-5xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-2 : 5 : 1) + (2 : -5 : 1) - D_\infty\) \((x - 2z) (x + 2z)\) \(=\) \(0,\) \(2y\) \(=\) \(-5xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \((x - 2z) (x + 2z)\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 - 9xz^2\) \(0\) \(2\)

2-torsion field: 4.2.1104332.3

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)   (lower bound)
2-Selmer rank:\(3\)
Regulator: \( 1.0 \)
Real period: \( 1.196177 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 4.784709 \)
Analytic order of Ш: \( 16 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(276083\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 436 T + 276083 T^{2} )\)

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?
\(2\) 2.30.3 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);