Properties

Label 864.a.442368.1
Conductor 864
Discriminant 442368
Sato-Tate group $N(G_{1,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{CM} \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 0, 6, 0, -4, 0, 1], R![]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 0, 6, 0, -4, 0, 1]), R([]))
 

$y^2 = x^6 - 4x^4 + 6x^2 - 3$

Invariants

magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(864,2),R![1]>*])); Factorization($1);
 
\( N \)  =  \( 864 \)  =  \( 2^{5} \cdot 3^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(442368\)  =  \( 2^{14} \cdot 3^{3} \)

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);
 

\( I_2 \)  =  \(17664\)  =  \( 2^{8} \cdot 3 \cdot 23 \)
\( I_4 \)  =  \(46080\)  =  \( 2^{10} \cdot 3^{2} \cdot 5 \)
\( I_6 \)  =  \(232095744\)  =  \( 2^{15} \cdot 3^{2} \cdot 787 \)
\( I_{10} \)  =  \(1811939328\)  =  \( 2^{26} \cdot 3^{3} \)
\( J_2 \)  =  \(2208\)  =  \( 2^{5} \cdot 3 \cdot 23 \)
\( J_4 \)  =  \(202656\)  =  \( 2^{5} \cdot 3 \cdot 2111 \)
\( J_6 \)  =  \(24809472\)  =  \( 2^{12} \cdot 3^{2} \cdot 673 \)
\( J_8 \)  =  \(3427464960\)  =  \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 297523 \)
\( J_{10} \)  =  \(442368\)  =  \( 2^{14} \cdot 3^{3} \)
\( g_1 \)  =  \(118634674176\)
\( g_2 \)  =  \(4931431104\)
\( g_3 \)  =  \(273421056\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(V_4 \) (GAP id : [4,2])

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

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

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

Number of rational Weierstrass points: \(2\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(1\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 9.0714829523350048361573285333

Tamagawa numbers: 6 (p = 2), 1 (p = 3)

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

Torsion: \(\Z/{12}\Z\)

2-torsion field: 4.0.432.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(G_{1,3})$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 24.a5
  Elliptic curve 36.a4

Endomorphisms

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

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(4\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)