Properties

Label 800.a.409600.1
Conductor 800
Discriminant -409600
Mordell-Weil group \(\Z/{24}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  =  \(800\) = \( 2^{5} \cdot 5^{2} \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(800,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-409600\) = \( - 2^{14} \cdot 5^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-3840\) =  \( - 2^{8} \cdot 3 \cdot 5 \)
\( I_4 \)  = \(316416\) =  \( 2^{10} \cdot 3 \cdot 103 \)
\( I_6 \)  = \(-487882752\) =  \( - 2^{15} \cdot 3 \cdot 7 \cdot 709 \)
\( I_{10} \)  = \(-1677721600\) =  \( - 2^{26} \cdot 5^{2} \)
\( J_2 \)  = \(-480\) =  \( - 2^{5} \cdot 3 \cdot 5 \)
\( J_4 \)  = \(6304\) =  \( 2^{5} \cdot 197 \)
\( J_6 \)  = \(151552\) =  \( 2^{12} \cdot 37 \)
\( J_8 \)  = \(-28121344\) =  \( - 2^{8} \cdot 109849 \)
\( J_{10} \)  = \(-409600\) =  \( - 2^{14} \cdot 5^{2} \)
\( g_1 \)  = \(62208000\)
\( g_2 \)  = \(1702080\)
\( g_3 \)  = \(-85248\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,0,1],C![1,1,0]];
 

Number of rational Weierstrass points: \(2\)

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

This curve is locally solvable everywhere.

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: \(\Z/{24}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 + xz - 2z^2\) \(=\) \(0,\) \(4y\) \(=\) \(3xz^2 - 2z^3\) \(0\) \(24\)

2-torsion field: 4.0.320.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 16.77015 \)
Tamagawa product: \( 12 \)
Torsion order:\( 24 \)
Leading coefficient: \( 0.349378 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(14\) \(5\) \(12\) \(1\)
\(5\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 40.a3
  Elliptic curve 20.a3

Endomorphisms of the Jacobian

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 \(\Q\).