Properties

Label 360.a.6480.1
Conductor 360
Discriminant 6480
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

Learn more about

Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, 0, 7, 0, -3], R![0, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, 0, 7, 0, -3]), R([0, 1, 0, 1]))

$y^2 + (x^3 + x)y = -3x^4 + 7x^2 - 5$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 360 \)  =  \( 2^{3} \cdot 3^{2} \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(6480\)  =  \( 2^{4} \cdot 3^{4} \cdot 5 \)

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);

\( I_2 \)  =  \(9440\)  =  \( 2^{5} \cdot 5 \cdot 59 \)
\( I_4 \)  =  \(191872\)  =  \( 2^{7} \cdot 1499 \)
\( I_6 \)  =  \(579060480\)  =  \( 2^{8} \cdot 3 \cdot 5 \cdot 150797 \)
\( I_{10} \)  =  \(26542080\)  =  \( 2^{16} \cdot 3^{4} \cdot 5 \)
\( J_2 \)  =  \(1180\)  =  \( 2^{2} \cdot 5 \cdot 59 \)
\( J_4 \)  =  \(56018\)  =  \( 2 \cdot 37 \cdot 757 \)
\( J_6 \)  =  \(3453120\)  =  \( 2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 109 \)
\( J_8 \)  =  \(234166319\)  =  \( 3299 \cdot 70981 \)
\( J_{10} \)  =  \(6480\)  =  \( 2^{4} \cdot 3^{4} \cdot 5 \)
\( g_1 \)  =  \(28596971960000/81\)
\( g_2 \)  =  \(1150492082200/81\)
\( g_3 \)  =  \(6677950400/9\)
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![-2,5,1],C![-1,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-5,1]];

All rational points: (-2 : 5 : 1), (-1 : 1 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (2 : -5 : 1)

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

Number of rational Weierstrass points: \(4\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(3\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

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

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

Torsion: \(\Z/{2}\Z \times \Z/{2}\Z \times \Z/{8}\Z\)

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

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 15.a6
  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 \(\Q\).