Properties

Label 588.a.18816.1
Conductor 588
Discriminant -18816
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![8, 12, 5, 0, 1, 1], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([8, 12, 5, 0, 1, 1]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = x^5 + x^4 + 5x^2 + 12x + 8$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 588 \)  =  \( 2^{2} \cdot 3 \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-18816\)  =  \( -1 \cdot 2^{7} \cdot 3 \cdot 7^{2} \)

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);

\( I_2 \)  =  \(-1496\)  =  \( -1 \cdot 2^{3} \cdot 11 \cdot 17 \)
\( I_4 \)  =  \(46180\)  =  \( 2^{2} \cdot 5 \cdot 2309 \)
\( I_6 \)  =  \(-23222296\)  =  \( -1 \cdot 2^{3} \cdot 2902787 \)
\( I_{10} \)  =  \(-77070336\)  =  \( -1 \cdot 2^{19} \cdot 3 \cdot 7^{2} \)
\( J_2 \)  =  \(-187\)  =  \( -1 \cdot 11 \cdot 17 \)
\( J_4 \)  =  \(976\)  =  \( 2^{4} \cdot 61 \)
\( J_6 \)  =  \(192\)  =  \( 2^{6} \cdot 3 \)
\( J_8 \)  =  \(-247120\)  =  \( -1 \cdot 2^{4} \cdot 5 \cdot 3089 \)
\( J_{10} \)  =  \(-18816\)  =  \( -1 \cdot 2^{7} \cdot 3 \cdot 7^{2} \)
\( g_1 \)  =  \(228669389707/18816\)
\( g_2 \)  =  \(398891383/1176\)
\( g_3 \)  =  \(-34969/98\)
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![-2,3,1],C![-2,4,1],C![-1,-1,1],C![-1,1,1],C![1,-1,0],C![1,0,0]];

All rational points: (-2 : 3 : 1), (-2 : 4 : 1), (-1 : -1 : 1), (-1 : 1 : 1), (1 : -1 : 0), (1 : 0 : 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: \(1\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 8 (p = 2), 1 (p = 3), 1 (p = 7)

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

Torsion: \(\Z/{24}\Z\)

2-torsion field: 8.0.12446784.1

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 14.a5
  Elliptic curve 42.a5

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 \(\Q\).