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Properties

Label 8649.a.77841.1
Conductor $8649$
Discriminant $77841$
Mordell-Weil group \(\Z \oplus \Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\mathsf{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = x^3 + x^2 - 2x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^3z^3 + x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^4 + 6x^3 + 5x^2 - 6x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 1, 1]), R([1, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 1, 1], R![1, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, -6, 5, 6, 2, 0, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(92\) \(=\)  \( 2^{2} \cdot 23 \)
\( I_4 \)  \(=\) \(17689\) \(=\)  \( 7^{2} \cdot 19^{2} \)
\( I_6 \)  \(=\) \(603507\) \(=\)  \( 3 \cdot 37 \cdot 5437 \)
\( I_{10} \)  \(=\) \(-9963648\) \(=\)  \( - 2^{7} \cdot 3^{4} \cdot 31^{2} \)
\( J_2 \)  \(=\) \(23\) \(=\)  \( 23 \)
\( J_4 \)  \(=\) \(-715\) \(=\)  \( - 5 \cdot 11 \cdot 13 \)
\( J_6 \)  \(=\) \(-3645\) \(=\)  \( - 3^{6} \cdot 5 \)
\( J_8 \)  \(=\) \(-148765\) \(=\)  \( - 5 \cdot 29753 \)
\( J_{10} \)  \(=\) \(-77841\) \(=\)  \( - 3^{4} \cdot 31^{2} \)
\( g_1 \)  \(=\) \(-6436343/77841\)
\( g_2 \)  \(=\) \(8699405/77841\)
\( g_3 \)  \(=\) \(23805/961\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

Copy content sage:C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
Copy content magma:IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-1 : -1 : 1)\)
\((-2 : 0 : 1)\) \((-1 : 2 : 1)\) \((1 : -3 : 1)\) \((1 : -5 : 2)\) \((1 : -8 : 2)\) \((4 : 8 : 3)\)
\((-2 : 9 : 1)\) \((4 : -135 : 3)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-1 : -1 : 1)\)
\((-2 : 0 : 1)\) \((-1 : 2 : 1)\) \((1 : -3 : 1)\) \((1 : -5 : 2)\) \((1 : -8 : 2)\) \((4 : 8 : 3)\)
\((-2 : 9 : 1)\) \((4 : -135 : 3)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -3 : 1)\) \((-1 : 3 : 1)\)
\((1 : -3 : 1)\) \((1 : 3 : 1)\) \((1 : -3 : 2)\) \((1 : 3 : 2)\) \((-2 : -9 : 1)\) \((-2 : 9 : 1)\)
\((4 : -143 : 3)\) \((4 : 143 : 3)\)

Copy content 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]]; // minimal model
 
Copy content magma:[C![-2,-9,1],C![-2,9,1],C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-3,2],C![1,3,2],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,3,1],C![4,-143,3],C![4,143,3]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable everywhere.

Copy content 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);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.066384\) \(\infty\)
\((-1 : 2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.070872\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.066384\) \(\infty\)
\((-1 : 2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.070872\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -3 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - z^3\) \(0.066384\) \(\infty\)
\((-1 : 3 : 1) + (1 : 3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - xz^2 + 3z^3\) \(0.070872\) \(\infty\)

2-torsion field: 3.1.31.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.004684 \)
Real period: \( 18.14230 \)
Tamagawa product: \( 4 \)
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 Root number L-factor Cluster picture Tame reduction?
\(3\) \(2\) \(4\) \(4\) \(1\) \(( 1 + T )^{2}\) yes
\(31\) \(2\) \(2\) \(1\) \(1\) \(( 1 + T )^{2}\) yes

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.120.2 no
\(3\) 3.432.4 no

Sato-Tate group

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

Decomposition of the Jacobian

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

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

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 \(\Q\).

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);