Properties

Label 548740.a.548740.1
Conductor $548740$
Discriminant $-548740$
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

$y^2 + (x^3 + 1)y = -4x^6 + 2x^5 - 14x^4 + 18x^3 - 20x^2 + 25x - 23$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -4x^6 + 2x^5z - 14x^4z^2 + 18x^3z^3 - 20x^2z^4 + 25xz^5 - 23z^6$ (dehomogenize, simplify)
$y^2 = -15x^6 + 8x^5 - 56x^4 + 74x^3 - 80x^2 + 100x - 91$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-23, 25, -20, 18, -14, 2, -4]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-23, 25, -20, 18, -14, 2, -4], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-91, 100, -80, 74, -56, 8, -15]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(548740\) \(=\) \( 2^{2} \cdot 5 \cdot 27437 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-548740\) \(=\) \( - 2^{2} \cdot 5 \cdot 27437 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(167212\) \(=\)  \( 2^{2} \cdot 17 \cdot 2459 \)
\( I_4 \)  \(=\) \(824940097\) \(=\)  \( 12973 \cdot 63589 \)
\( I_6 \)  \(=\) \(38082033914927\) \(=\)  \( 9533 \cdot 3994758619 \)
\( I_{10} \)  \(=\) \(70238720\) \(=\)  \( 2^{9} \cdot 5 \cdot 27437 \)
\( J_2 \)  \(=\) \(41803\) \(=\)  \( 17 \cdot 2459 \)
\( J_4 \)  \(=\) \(38439613\) \(=\)  \( 71 \cdot 109 \cdot 4967 \)
\( J_6 \)  \(=\) \(39312521235\) \(=\)  \( 3^{3} \cdot 5 \cdot 13 \cdot 19 \cdot 1178963 \)
\( J_8 \)  \(=\) \(41444369399234\) \(=\)  \( 2 \cdot 83^{3} \cdot 36241091 \)
\( J_{10} \)  \(=\) \(548740\) \(=\)  \( 2^{2} \cdot 5 \cdot 27437 \)
\( g_1 \)  \(=\) \(127654829703532651729243/548740\)
\( g_2 \)  \(=\) \(2808027502126164181351/548740\)
\( g_3 \)  \(=\) \(13739653907355965823/109748\)

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
\(D_0 - D_\infty\) \(3x^2 + 2xz + 7z^2\) \(=\) \(0,\) \(18y\) \(=\) \(17xz^2 - 23z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(3x^2 + 2xz + 7z^2\) \(=\) \(0,\) \(18y\) \(=\) \(17xz^2 - 23z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(3x^2 + 2xz + 7z^2\) \(=\) \(0,\) \(18y\) \(=\) \(x^3 + 34xz^2 - 45z^3\) \(0\) \(2\)

2-torsion field: 6.4.24089247008000.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(4\)
Regulator: \( 1 \)
Real period: \( 1.286350 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.572701 \)
Analytic order of Ш: \( 8 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(2\) \(1\) \(1 + T^{2}\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(27437\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 186 T + 27437 T^{2} )\)

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);