Properties

Label 960.a.983040.1
Conductor 960
Discriminant -983040
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![0, 1, -2, -1, -2, 1], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -2, -1, -2, 1]), R([]))

$y^2 = x^5 - 2x^4 - x^3 - 2x^2 + x$

Invariants

magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(960,2),R![1]>*])); Factorization($1);
\( N \)  =  \( 960 \)  =  \( 2^{6} \cdot 3 \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-983040\)  =  \( -1 \cdot 2^{16} \cdot 3 \cdot 5 \)

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);

\( I_2 \)  =  \(-288\)  =  \( -1 \cdot 2^{5} \cdot 3^{2} \)
\( I_4 \)  =  \(33792\)  =  \( 2^{10} \cdot 3 \cdot 11 \)
\( I_6 \)  =  \(-21823488\)  =  \( -1 \cdot 2^{16} \cdot 3^{2} \cdot 37 \)
\( I_{10} \)  =  \(-4026531840\)  =  \( -1 \cdot 2^{28} \cdot 3 \cdot 5 \)
\( J_2 \)  =  \(-36\)  =  \( -1 \cdot 2^{2} \cdot 3^{2} \)
\( J_4 \)  =  \(-298\)  =  \( -1 \cdot 2 \cdot 149 \)
\( J_6 \)  =  \(34260\)  =  \( 2^{2} \cdot 3 \cdot 5 \cdot 571 \)
\( J_8 \)  =  \(-330541\)  =  \( -1 \cdot 43 \cdot 7687 \)
\( J_{10} \)  =  \(-983040\)  =  \( -1 \cdot 2^{16} \cdot 3 \cdot 5 \)
\( g_1 \)  =  \(19683/320\)
\( g_2 \)  =  \(-36207/2560\)
\( g_3 \)  =  \(-46251/1024\)
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![0,0,1],C![1,0,0]];

All rational points: (0 : 0 : 1), (1 : 0 : 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: \(2\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 4 (p = 2), 1 (p = 3), 1 (p = 5)

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

Torsion: \(\Z/{2}\Z \times \Z/{4}\Z\)

2-torsion field: \(\Q(\sqrt{-3}, \sqrt{5})\)

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 40.a2
  Elliptic curve 24.a4

Endomorphisms

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

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

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