Properties

Label 8649.a.233523.1
Conductor 8649
Discriminant 233523
Mordell-Weil group \(\Z \times \Z\)
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

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Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-70, -47, 7, 3, -3], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-70, -47, 7, 3, -3]), R([1, 1, 0, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-70, -47, 7, 3, -3], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-279, -186, 29, 14, -10, 0, 1]))
 

$y^2 + (x^3 + x + 1)y = -3x^4 + 3x^3 + 7x^2 - 47x - 70$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -3x^4z^2 + 3x^3z^3 + 7x^2z^4 - 47xz^5 - 70z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 10x^4 + 14x^3 + 29x^2 - 186x - 279$ (minimize, homogenize)

Invariants

\( N \)  =  \(8649\) = \( 3^{2} \cdot 31^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(233523\) = \( 3^{5} \cdot 31^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

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);
 

\( I_2 \)  = \(72776\) =  \( 2^{3} \cdot 11 \cdot 827 \)
\( I_4 \)  = \(-4565084\) =  \( - 2^{2} \cdot 1141271 \)
\( I_6 \)  = \(-110566728408\) =  \( - 2^{3} \cdot 3 \cdot 4606947017 \)
\( I_{10} \)  = \(956510208\) =  \( 2^{12} \cdot 3^{5} \cdot 31^{2} \)
\( J_2 \)  = \(9097\) =  \( 11 \cdot 827 \)
\( J_4 \)  = \(3495695\) =  \( 5 \cdot 7 \cdot 99877 \)
\( J_6 \)  = \(1814445117\) =  \( 3^{4} \cdot 29 \cdot 107 \cdot 7219 \)
\( J_8 \)  = \(1071530924081\) =  \( 23 \cdot 157 \cdot 296740771 \)
\( J_{10} \)  = \(233523\) =  \( 3^{5} \cdot 31^{2} \)
\( g_1 \)  = \(62300419867534985257/233523\)
\( g_2 \)  = \(2631649929116327735/233523\)
\( g_3 \)  = \(1853767256362813/2883\)

Automorphism group

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

Rational points

magma: [C![-7,189,3],C![-7,190,3],C![-3,10,1],C![-3,19,1],C![-2,4,1],C![-2,5,1],C![1,-1,0],C![1,0,0]];
 

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-2 : 4 : 1)\) \((-2 : 5 : 1)\) \((-3 : 10 : 1)\) \((-3 : 19 : 1)\)
\((-7 : 189 : 3)\) \((-7 : 190 : 3)\)

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

Number of rational Weierstrass points: \(0\)

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.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z \times \Z\)

Generator Height Order
\(z (3x + 7z)\) \(=\) \(0,\) \(3y\) \(=\) \(-3x^3 - 17z^3\) \(0.132768\) \(\infty\)
\(z (x + 3z)\) \(=\) \(0,\) \(y\) \(=\) \(10z^3\) \(0.141745\) \(\infty\)

2-torsion field: 6.2.184512.1

BSD invariants

Analytic rank: \(2\)   (upper bound)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.018738 \)
Real period: \( 9.071150 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.339964 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(5\) \(2\) \(2\) \(( 1 + T )^{2}\)
\(31\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\sqrt{5}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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