Properties

Label 8649.a.77841.1
Conductor 8649
Discriminant 77841
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 1, 1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 1, 1]), R([1, 1, 0, 1]))
 

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 8649 \)  =  \( 3^{2} \cdot 31^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(77841\)  =  \( 3^{4} \cdot 31^{2} \)

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);
 

\( I_2 \)  =  \(-184\)  =  \( -1 \cdot 2^{3} \cdot 23 \)
\( I_4 \)  =  \(70756\)  =  \( 2^{2} \cdot 7^{2} \cdot 19^{2} \)
\( I_6 \)  =  \(-4828056\)  =  \( -1 \cdot 2^{3} \cdot 3 \cdot 37 \cdot 5437 \)
\( I_{10} \)  =  \(318836736\)  =  \( 2^{12} \cdot 3^{4} \cdot 31^{2} \)
\( J_2 \)  =  \(-23\)  =  \( -1 \cdot 23 \)
\( J_4 \)  =  \(-715\)  =  \( -1 \cdot 5 \cdot 11 \cdot 13 \)
\( J_6 \)  =  \(3645\)  =  \( 3^{6} \cdot 5 \)
\( J_8 \)  =  \(-148765\)  =  \( -1 \cdot 5 \cdot 29753 \)
\( J_{10} \)  =  \(77841\)  =  \( 3^{4} \cdot 31^{2} \)
\( g_1 \)  =  \(-6436343/77841\)
\( g_2 \)  =  \(8699405/77841\)
\( g_3 \)  =  \(23805/961\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_2 \) (GAP id : [2,1])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(C_2 \) (GAP id : [2,1])

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![-2,0,1],C![-2,9,1],C![-1,-1,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-8,2],C![1,-5,2],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![4,-135,3],C![4,8,3]];
 

Known rational points: (-2 : 0 : 1), (-2 : 9 : 1), (-1 : -1 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -8 : 2), (1 : -5 : 2), (1 : -3 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (4 : -135 : 3), (4 : 8 : 3)

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank*: \(2\)

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

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.00468469737266

Real period: 18.142300326302809119878028598

Tamagawa numbers: 4 (p = 3), 1 (p = 31)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 3.1.31.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition

Simple over \(\overline{\Q}\)

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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