Properties

Label 676.b.17576.1
Conductor 676
Discriminant -17576
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

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Show commands for: Magma / SageMath

Minimal equation

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

$y^2 + (x^2 + x)y = -x^6 + 3x^5 - 6x^4 + 6x^3 - 6x^2 + 3x - 1$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 676 \)  =  \( 2^{2} \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-17576\)  =  \( -1 \cdot 2^{3} \cdot 13^{3} \)

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);

\( I_2 \)  =  \(-2488\)  =  \( -1 \cdot 2^{3} \cdot 311 \)
\( I_4 \)  =  \(4996\)  =  \( 2^{2} \cdot 1249 \)
\( I_6 \)  =  \(-1033336\)  =  \( -1 \cdot 2^{3} \cdot 37 \cdot 3491 \)
\( I_{10} \)  =  \(-71991296\)  =  \( -1 \cdot 2^{15} \cdot 13^{3} \)
\( J_2 \)  =  \(-311\)  =  \( -1 \cdot 311 \)
\( J_4 \)  =  \(3978\)  =  \( 2 \cdot 3^{2} \cdot 13 \cdot 17 \)
\( J_6 \)  =  \(-72332\)  =  \( -1 \cdot 2^{2} \cdot 13^{2} \cdot 107 \)
\( J_8 \)  =  \(1667692\)  =  \( 2^{2} \cdot 13^{2} \cdot 2467 \)
\( J_{10} \)  =  \(-17576\)  =  \( -1 \cdot 2^{3} \cdot 13^{3} \)
\( g_1 \)  =  \(2909390022551/17576\)
\( g_2 \)  =  \(4602275343/676\)
\( g_3 \)  =  \(10349147/26\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(D_6 \) (GAP id : [12,4])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(D_6 \) (GAP id : [12,4])

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 except over $\R$, $\Q_{2}$, and $\Q_{11}$.

magma: [];

There are no rational points.

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: \(0\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 1 (p = 2), 3 (p = 13)

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

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

2-torsion field: 3.1.104.1

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 26.a2

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an Eichler order of index \(3\) 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 \(\Q\).