Properties

Label 576.b.147456.1
Conductor 576
Discriminant -147456
Sato-Tate group $E_1$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands for: Magma / SageMath

Minimal equation

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

$y^2 = x^6 + 2x^4 + 2x^2 + 1$

Invariants

magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(576,2),R![1]>*])); Factorization($1);
 
\( N \)  =  \( 576 \)  =  \( 2^{6} \cdot 3^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-147456\)  =  \( -1 \cdot 2^{14} \cdot 3^{2} \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  =  \(-4864\)  =  \( -1 \cdot 2^{8} \cdot 19 \)
\( I_4 \)  =  \(111616\)  =  \( 2^{10} \cdot 109 \)
\( I_6 \)  =  \(-179208192\)  =  \( -1 \cdot 2^{15} \cdot 3 \cdot 1823 \)
\( I_{10} \)  =  \(-603979776\)  =  \( -1 \cdot 2^{26} \cdot 3^{2} \)
\( J_2 \)  =  \(-608\)  =  \( -1 \cdot 2^{5} \cdot 19 \)
\( J_4 \)  =  \(14240\)  =  \( 2^{5} \cdot 5 \cdot 89 \)
\( J_6 \)  =  \(-405504\)  =  \( -1 \cdot 2^{12} \cdot 3^{2} \cdot 11 \)
\( J_8 \)  =  \(10942208\)  =  \( 2^{8} \cdot 42743 \)
\( J_{10} \)  =  \(-147456\)  =  \( -1 \cdot 2^{14} \cdot 3^{2} \)
\( g_1 \)  =  \(5071050752/9\)
\( g_2 \)  =  \(195344320/9\)
\( g_3 \)  =  \(1016576\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(D_4 \) (GAP id : [8,3])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(D_4 \) (GAP id : [8,3])

Rational points

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);
 

This curve is locally solvable everywhere.

magma: [C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0]];
 

All rational points: (0 : -1 : 1), (0 : 1 : 1), (1 : -1 : 0), (1 : 1 : 0)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
 

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 9.3011194758216110390641169009

Tamagawa numbers: 8 (p = 2), 1 (p = 3)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
 

Torsion: \(\Z/{4}\Z \times \Z/{4}\Z\)

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

Sato-Tate group

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

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the square of the elliptic curve:
  Elliptic curve 24.a4

Endomorphisms

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 \(\Q\).