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Properties

Label 274688.a.274688.1
Conductor $274688$
Discriminant $-274688$
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

$y^2 + x^2y = x^5 - 32x^4 - 80x^3 - 66x^2 - 20x - 2$ (homogenize, simplify)
$y^2 + x^2zy = x^5z - 32x^4z^2 - 80x^3z^3 - 66x^2z^4 - 20xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 127x^4 - 320x^3 - 264x^2 - 80x - 8$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, -20, -66, -80, -32, 1]), R([0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, -20, -66, -80, -32, 1], R![0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-8, -80, -264, -320, -127, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(274688\) \(=\) \( 2^{8} \cdot 29 \cdot 37 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-274688\) \(=\) \( - 2^{8} \cdot 29 \cdot 37 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(8144\) \(=\)  \( 2^{4} \cdot 509 \)
\( I_4 \)  \(=\) \(6441502\) \(=\)  \( 2 \cdot 59 \cdot 79 \cdot 691 \)
\( I_6 \)  \(=\) \(12034879778\) \(=\)  \( 2 \cdot 359 \cdot 809 \cdot 20719 \)
\( I_{10} \)  \(=\) \(-34336\) \(=\)  \( - 2^{5} \cdot 29 \cdot 37 \)
\( J_2 \)  \(=\) \(8144\) \(=\)  \( 2^{4} \cdot 509 \)
\( J_4 \)  \(=\) \(-1530804\) \(=\)  \( - 2^{2} \cdot 3 \cdot 11 \cdot 11597 \)
\( J_6 \)  \(=\) \(267412496\) \(=\)  \( 2^{4} \cdot 13 \cdot 41 \cdot 31357 \)
\( J_8 \)  \(=\) \(-41388379748\) \(=\)  \( - 2^{2} \cdot 563 \cdot 1879 \cdot 9781 \)
\( J_{10} \)  \(=\) \(-274688\) \(=\)  \( - 2^{8} \cdot 29 \cdot 37 \)
\( g_1 \)  \(=\) \(-139942252386504704/1073\)
\( g_2 \)  \(=\) \(3229928570273856/1073\)
\( g_3 \)  \(=\) \(-69281496876176/1073\)

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

magma: [C![-1,-2,4],C![1,0,0]]; // minimal model
 
magma: [C![-1,0,4],C![1,0,0]]; // 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
\((-1 : -2 : 4) - (1 : 0 : 0)\) \(4x + z\) \(=\) \(0,\) \(32y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-1 : -2 : 4) - (1 : 0 : 0)\) \(4x + z\) \(=\) \(0,\) \(32y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x + z\) \(=\) \(0,\) \(32y\) \(=\) \(x^2z - 2z^3\) \(0\) \(2\)

2-torsion field: 4.2.1098752.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)   (lower bound)
2-Selmer rank:\(3\)
Regulator: \( 1.0 \)
Real period: \( 0.783612 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 3.134449 \)
Analytic order of Ш: \( 16 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(8\) \(1\) \(1 - T\)
\(29\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 29 T^{2} )\)
\(37\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 2 T + 37 T^{2} )\)

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);