Properties

Label 8450.c.84500.1
Conductor 8450
Discriminant 84500
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

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Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -5, 10, -5], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -5, 10, -5]), R([1, 0, 0, 1]))
 

$y^2 + (x^3 + 1)y = -5x^4 + 10x^3 - 5x^2$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 8450 \)  =  \( 2 \cdot 5^{2} \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(84500\)  =  \( 2^{2} \cdot 5^{3} \cdot 13^{2} \)

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);
 

\( I_2 \)  =  \(3944\)  =  \( 2^{3} \cdot 17 \cdot 29 \)
\( I_4 \)  =  \(243556\)  =  \( 2^{2} \cdot 60889 \)
\( I_6 \)  =  \(286158056\)  =  \( 2^{3} \cdot 35769757 \)
\( I_{10} \)  =  \(346112000\)  =  \( 2^{14} \cdot 5^{3} \cdot 13^{2} \)
\( J_2 \)  =  \(493\)  =  \( 17 \cdot 29 \)
\( J_4 \)  =  \(7590\)  =  \( 2 \cdot 3 \cdot 5 \cdot 11 \cdot 23 \)
\( J_6 \)  =  \(128000\)  =  \( 2^{10} \cdot 5^{3} \)
\( J_8 \)  =  \(1373975\)  =  \( 5^{2} \cdot 54959 \)
\( J_{10} \)  =  \(84500\)  =  \( 2^{2} \cdot 5^{3} \cdot 13^{2} \)
\( g_1 \)  =  \(29122898485693/84500\)
\( g_2 \)  =  \(90945776163/8450\)
\( g_3 \)  =  \(62220544/169\)
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,-1,1],C![0,0,1],C![1,-18,3],C![1,-10,3],C![1,-5,2],C![1,-4,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-5,1],C![2,-4,1],C![3,-18,1],C![3,-10,1]];
 

Known rational points: (0 : -1 : 1), (0 : 0 : 1), (1 : -18 : 3), (1 : -10 : 3), (1 : -5 : 2), (1 : -4 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (2 : -5 : 1), (2 : -4 : 1), (3 : -18 : 1), (3 : -10 : 1)

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: \(3\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.109881447913

Real period: 20.369950202679124490548652016

Tamagawa numbers: 2 (p = 2), 3 (p = 5), 1 (p = 13)

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

Torsion: \(\Z/{6}\Z\)

2-torsion field: \(\Q(\sqrt{5}, \sqrt{13})\)

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 65.a1
  Elliptic curve 130.a2

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 \(\Q\).