Properties

Label 8664.c.658464.1
Conductor $8664$
Discriminant $658464$
Mordell-Weil group \(\Z/{10}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x + 1)y = -2x^5 + 4x^4 + 2x^3 + 2x^2$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = -2x^5z + 4x^4z^2 + 2x^3z^3 + 2x^2z^4$ (dehomogenize, simplify)
$y^2 = -8x^5 + 16x^4 + 8x^3 + 9x^2 + 2x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); Cmin = HyperellipticCurve(R([0, 0, 2, 2, 4, -2]), R([1, 1])) # minimal equation
 
Copy content magma:R<x> := PolynomialRing(Rationals()); Cmin := HyperellipticCurve(R![0, 0, 2, 2, 4, -2], R![1, 1]); // minimal equation
 
Copy content sage:Csim = HyperellipticCurve(R([1, 2, 9, 8, 16, -8])); Csim # simplified equation
 
Copy content magma:Csim, pi := SimplifiedModel(Cmin); Csim; // simplified equation
 

Invariants

Conductor: \( N \)  \(=\)  \(8664\) \(=\) \( 2^{3} \cdot 3 \cdot 19^{2} \)
Copy content magma:Conductor(LSeries(Cmin)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(658464\) \(=\) \( 2^{5} \cdot 3 \cdot 19^{3} \)
Copy content magma:Discriminant(Cmin); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(640\) \(=\)  \( 2^{7} \cdot 5 \)
\( I_4 \)  \(=\) \(36448\) \(=\)  \( 2^{5} \cdot 17 \cdot 67 \)
\( I_6 \)  \(=\) \(5569220\) \(=\)  \( 2^{2} \cdot 5 \cdot 23 \cdot 12107 \)
\( I_{10} \)  \(=\) \(-2633856\) \(=\)  \( - 2^{7} \cdot 3 \cdot 19^{3} \)
\( J_2 \)  \(=\) \(320\) \(=\)  \( 2^{6} \cdot 5 \)
\( J_4 \)  \(=\) \(-1808\) \(=\)  \( - 2^{4} \cdot 113 \)
\( J_6 \)  \(=\) \(-2980\) \(=\)  \( - 2^{2} \cdot 5 \cdot 149 \)
\( J_8 \)  \(=\) \(-1055616\) \(=\)  \( - 2^{7} \cdot 3 \cdot 2749 \)
\( J_{10} \)  \(=\) \(-658464\) \(=\)  \( - 2^{5} \cdot 3 \cdot 19^{3} \)
\( g_1 \)  \(=\) \(-104857600000/20577\)
\( g_2 \)  \(=\) \(1851392000/20577\)
\( g_3 \)  \(=\) \(9536000/20577\)

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

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

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)
All points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1)\)

Copy content magma:[Cmin![0,-1,1],Cmin![0,0,1],Cmin![1,0,0]]; // minimal model
 
Copy content magma:[Csim![0,-1,1],Csim![0,1,1],Csim![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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

This curve is locally solvable everywhere.

Copy content magma:f,h:=HyperellipticPolynomials(Cmin); 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: \(\Z/{10}\Z\)

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

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2\) \(0\) \(10\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2\) \(0\) \(10\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(5xz^2 + z^3\) \(0\) \(10\)

2-torsion field: 6.0.623808.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 10.14385 \)
Tamagawa product: \( 10 \)
Torsion order:\( 10 \)
Leading coefficient: \( 1.014385 \)
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?
\(2\) \(3\) \(5\) \(5\) \(1^*\) \(1 - T\) yes
\(3\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + T + 3 T^{2} )\) yes
\(19\) \(2\) \(3\) \(2\) \(1\) \(( 1 + T )^{2}\) 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?
\(2\) 2.60.1 yes
\(3\) 3.80.4 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

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 38.b
  Elliptic curve isogeny class 228.b

Copy content magma:HeuristicDecompositionFactors(Cmin);
 

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(3\) 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 \(\Q\).

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

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

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