Properties

Label 2304.b.147456.1
Conductor $2304$
Discriminant $-147456$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Sato-Tate group $E_1$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 = -x^6 - 2x^4 - 2x^2 - 1$ (homogenize, simplify)
$y^2 = -x^6 - 2x^4z^2 - 2x^2z^4 - z^6$ (dehomogenize, simplify)
$y^2 = -x^6 - 2x^4 - 2x^2 - 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 0, -2, 0, -2, 0, -1]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 0, -2, 0, -2, 0, -1], R![]);
 
sage: X = HyperellipticCurve(R([-1, 0, -2, 0, -2, 0, -1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2304\) \(=\) \( 2^{8} \cdot 3^{2} \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(2304,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-147456\) \(=\) \( - 2^{14} \cdot 3^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(152\) \(=\)  \( 2^{3} \cdot 19 \)
\( I_4 \)  \(=\) \(109\) \(=\)  \( 109 \)
\( I_6 \)  \(=\) \(5469\) \(=\)  \( 3 \cdot 1823 \)
\( I_{10} \)  \(=\) \(18\) \(=\)  \( 2 \cdot 3^{2} \)
\( J_2 \)  \(=\) \(608\) \(=\)  \( 2^{5} \cdot 19 \)
\( J_4 \)  \(=\) \(14240\) \(=\)  \( 2^{5} \cdot 5 \cdot 89 \)
\( J_6 \)  \(=\) \(405504\) \(=\)  \( 2^{12} \cdot 3^{2} \cdot 11 \)
\( J_8 \)  \(=\) \(10942208\) \(=\)  \( 2^{8} \cdot 42743 \)
\( J_{10} \)  \(=\) \(147456\) \(=\)  \( 2^{14} \cdot 3^{2} \)
\( g_1 \)  \(=\) \(5071050752/9\)
\( g_2 \)  \(=\) \(195344320/9\)
\( g_3 \)  \(=\) \(1016576\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $D_4$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_4$
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 $\R$, $\Q_{2}$, 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: \(\Z/{2}\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: \(\Q(\zeta_{12})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 5.683508 \)
Tamagawa product: \( 2 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.710438 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(14\) \(2\) \(1\)
\(3\) \(2\) \(2\) \(1\) \(( 1 - T )^{2}\)

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?
\(2\) 2.180.7 yes
\(3\) 3.2160.25 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_1$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 48.a

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{}) \otimes \Q\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

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);