Properties

Label 644.a.659456.1
Conductor 644
Discriminant 659456
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-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, -13, 4, 51, 4, -13, -3], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -13, 4, 51, 4, -13, -3]), R([0, 1, 1]))
 

$y^2 + (x^2 + x)y = -3x^6 - 13x^5 + 4x^4 + 51x^3 + 4x^2 - 13x - 3$

Invariants

magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(644,2),R![1, 2, 1]>*])); Factorization($1);
 
\( N \)  =  \( 644 \)  =  \( 2^{2} \cdot 7 \cdot 23 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(659456\)  =  \( 2^{12} \cdot 7 \cdot 23 \)

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);
 

\( I_2 \)  =  \(323592\)  =  \( 2^{3} \cdot 3 \cdot 97 \cdot 139 \)
\( I_4 \)  =  \(4282649220\)  =  \( 2^{2} \cdot 3 \cdot 5 \cdot 23 \cdot 1487 \cdot 2087 \)
\( I_6 \)  =  \(368522127355272\)  =  \( 2^{3} \cdot 3 \cdot 11 \cdot 1395917149073 \)
\( I_{10} \)  =  \(2701131776\)  =  \( 2^{24} \cdot 7 \cdot 23 \)
\( J_2 \)  =  \(40449\)  =  \( 3 \cdot 97 \cdot 139 \)
\( J_4 \)  =  \(23560804\)  =  \( 2^{2} \cdot 199 \cdot 29599 \)
\( J_6 \)  =  \(14638854160\)  =  \( 2^{4} \cdot 5 \cdot 7 \cdot 23 \cdot 1136557 \)
\( J_8 \)  =  \(9253881697856\)  =  \( 2^{6} \cdot 54101 \cdot 2672629 \)
\( J_{10} \)  =  \(659456\)  =  \( 2^{12} \cdot 7 \cdot 23 \)
\( g_1 \)  =  \(108277681088425330677249/659456\)
\( g_2 \)  =  \(389810454818831018649/164864\)
\( g_3 \)  =  \(9297727292338785/256\)
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![-3,-3,1],C![-1,3,3]];
 

All rational points: (-3 : -3 : 1), (-1 : 3 : 3)

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: 0.87298469646885569477520610261

Tamagawa numbers: 1 (p = 2), 1 (p = 7), 1 (p = 23)

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

Torsion: \(\Z/{2}\Z\)

2-torsion field: 4.4.10304.1

Sato-Tate group

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

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 46.a1
  Elliptic curve 14.a1

Endomorphisms

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

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).