Properties

Label 679024.a.679024.1
Conductor $679024$
Discriminant $-679024$
Mordell-Weil group \(\Z/{3}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + y = -x^6 - 8x^4 - 18x^2 - 8$ (homogenize, simplify)
$y^2 + z^3y = -x^6 - 8x^4z^2 - 18x^2z^4 - 8z^6$ (dehomogenize, simplify)
$y^2 = -4x^6 - 32x^4 - 72x^2 - 31$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-8, 0, -18, 0, -8, 0, -1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-8, 0, -18, 0, -8, 0, -1], R![1]);
 
sage: X = HyperellipticCurve(R([-31, 0, -72, 0, -32, 0, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(679024\) \(=\) \( 2^{4} \cdot 31 \cdot 37^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-679024\) \(=\) \( - 2^{4} \cdot 31 \cdot 37^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(8328\) \(=\)  \( 2^{3} \cdot 3 \cdot 347 \)
\( I_4 \)  \(=\) \(352725\) \(=\)  \( 3 \cdot 5^{2} \cdot 4703 \)
\( I_6 \)  \(=\) \(909501207\) \(=\)  \( 3 \cdot 17^{2} \cdot 719 \cdot 1459 \)
\( I_{10} \)  \(=\) \(84878\) \(=\)  \( 2 \cdot 31 \cdot 37^{2} \)
\( J_2 \)  \(=\) \(8328\) \(=\)  \( 2^{3} \cdot 3 \cdot 347 \)
\( J_4 \)  \(=\) \(2654666\) \(=\)  \( 2 \cdot 7 \cdot 189619 \)
\( J_6 \)  \(=\) \(1072556352\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 31 \cdot 8581 \)
\( J_8 \)  \(=\) \(471249431975\) \(=\)  \( 5^{2} \cdot 7 \cdot 2692853897 \)
\( J_{10} \)  \(=\) \(679024\) \(=\)  \( 2^{4} \cdot 31 \cdot 37^{2} \)
\( g_1 \)  \(=\) \(2503707555146139648/42439\)
\( g_2 \)  \(=\) \(95832331547948352/42439\)
\( g_3 \)  \(=\) \(149975347108608/1369\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
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 $\R$ and $\Q_{3}$.

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: \(\Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(3\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(3\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(3\)

2-torsion field: 6.0.679024.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 2.993255 \)
Tamagawa product: \( 1 \)
Torsion order:\( 3 \)
Leading coefficient: \( 2.660671 \)
Analytic order of Ш: \( 8 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(4\) \(1\) \(1 + 2 T^{2}\)
\(31\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 31 T^{2} )\)
\(37\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.15.2 no
\(3\) 3.720.4 yes

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 18352.n
  Elliptic curve isogeny class 37.b

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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);