Properties

Label 720.b.116640.1
Conductor 720
Discriminant 116640
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![-90, 0, 39, 0, -6], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-90, 0, 39, 0, -6]), R([0, 1, 0, 1]))
 

$y^2 + (x^3 + x)y = -6x^4 + 39x^2 - 90$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 720 \)  =  \( 2^{4} \cdot 3^{2} \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(116640\)  =  \( 2^{5} \cdot 3^{6} \cdot 5 \)

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);
 

\( I_2 \)  =  \(141664\)  =  \( 2^{5} \cdot 19 \cdot 233 \)
\( I_4 \)  =  \(731008\)  =  \( 2^{7} \cdot 5711 \)
\( I_6 \)  =  \(34370557696\)  =  \( 2^{8} \cdot 6373 \cdot 21067 \)
\( I_{10} \)  =  \(477757440\)  =  \( 2^{17} \cdot 3^{6} \cdot 5 \)
\( J_2 \)  =  \(17708\)  =  \( 2^{2} \cdot 19 \cdot 233 \)
\( J_4 \)  =  \(13057938\)  =  \( 2 \cdot 3^{2} \cdot 17 \cdot 139 \cdot 307 \)
\( J_6 \)  =  \(12831384960\)  =  \( 2^{7} \cdot 3^{4} \cdot 5 \cdot 247519 \)
\( J_8 \)  =  \(14177105014959\)  =  \( 3^{4} \cdot 29 \cdot 6035378891 \)
\( J_{10} \)  =  \(116640\)  =  \( 2^{5} \cdot 3^{6} \cdot 5 \)
\( g_1 \)  =  \(54412363190235229024/3645\)
\( g_2 \)  =  \(251762275020280012/405\)
\( g_3 \)  =  \(310461362928064/9\)
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![-3,15,1],C![1,-1,0],C![1,0,0],C![3,-15,1]];
 

All rational points: (-3 : 15 : 1), (1 : -1 : 0), (1 : 0 : 0), (3 : -15 : 1)

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

Regulator: 1.0

Real period: 14.457057741326955436401008636

Tamagawa numbers: 2 (p = 2), 6 (p = 3), 1 (p = 5)

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

Torsion: \(\Z/{2}\Z \times \Z/{12}\Z\)

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

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 30.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 \(\Q\).