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Properties

Label 10000.a.160000.1
Conductor $10000$
Discriminant $-160000$
Mordell-Weil group \(\Z/{5}\Z\)
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

$y^2 + xy = x^6 - 2x^5 + 2x^4 - x^3 + 2x^2 - 3x + 2$ (homogenize, simplify)
$y^2 + xz^2y = x^6 - 2x^5z + 2x^4z^2 - x^3z^3 + 2x^2z^4 - 3xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = 4x^6 - 8x^5 + 8x^4 - 4x^3 + 9x^2 - 12x + 8$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -3, 2, -1, 2, -2, 1]), R([0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -3, 2, -1, 2, -2, 1], R![0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([8, -12, 9, -4, 8, -8, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10000\) \(=\) \( 2^{4} \cdot 5^{4} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-160000\) \(=\) \( - 2^{8} \cdot 5^{4} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(612\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 17 \)
\( I_4 \)  \(=\) \(11325\) \(=\)  \( 3 \cdot 5^{2} \cdot 151 \)
\( I_6 \)  \(=\) \(1916325\) \(=\)  \( 3^{3} \cdot 5^{2} \cdot 17 \cdot 167 \)
\( I_{10} \)  \(=\) \(20000\) \(=\)  \( 2^{5} \cdot 5^{4} \)
\( J_2 \)  \(=\) \(612\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 17 \)
\( J_4 \)  \(=\) \(8056\) \(=\)  \( 2^{3} \cdot 19 \cdot 53 \)
\( J_6 \)  \(=\) \(110704\) \(=\)  \( 2^{4} \cdot 11 \cdot 17 \cdot 37 \)
\( J_8 \)  \(=\) \(712928\) \(=\)  \( 2^{5} \cdot 22279 \)
\( J_{10} \)  \(=\) \(160000\) \(=\)  \( 2^{8} \cdot 5^{4} \)
\( g_1 \)  \(=\) \(335364543972/625\)
\( g_2 \)  \(=\) \(7213296078/625\)
\( g_3 \)  \(=\) \(161966871/625\)

  (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

All points: \((1 : -1 : 0),\, (1 : 1 : 0)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0)\)
All points: \((1 : -2 : 0),\, (1 : 2 : 0)\)

Copy content magma:[C![1,-1,0],C![1,1,0]]; // minimal model
 
Copy content magma:[C![1,-2,0],C![1,2,0]]; // 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: \(\Z/{5}\Z\)

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

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

2-torsion field: 6.0.640000.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 8.650087 \)
Tamagawa product: \( 5 \)
Torsion order:\( 5 \)
Leading coefficient: \( 1.730017 \)
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?
\(2\) \(4\) \(8\) \(5\) \(1^*\) \(1 - T\) no
\(5\) \(4\) \(4\) \(1\) \(1\) \(1\) 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?
\(2\) 2.60.2 no
\(3\) 3.1080.9 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 50.b
  Elliptic curve isogeny class 200.e

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(4\) 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(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

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