# Properties

 Label 800.a.409600.1 Conductor 800 Discriminant -409600 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

Show commands for: Magma / SageMath

## Minimal equation

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([]))

$y^2 = x^6 - 2x^2 + 1$

## Invariants

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

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-3840$$ = $$-1 \cdot 2^{8} \cdot 3 \cdot 5$$ $$I_4$$ = $$316416$$ = $$2^{10} \cdot 3 \cdot 103$$ $$I_6$$ = $$-487882752$$ = $$-1 \cdot 2^{15} \cdot 3 \cdot 7 \cdot 709$$ $$I_{10}$$ = $$-1677721600$$ = $$-1 \cdot 2^{26} \cdot 5^{2}$$ $$J_2$$ = $$-480$$ = $$-1 \cdot 2^{5} \cdot 3 \cdot 5$$ $$J_4$$ = $$6304$$ = $$2^{5} \cdot 197$$ $$J_6$$ = $$151552$$ = $$2^{12} \cdot 37$$ $$J_8$$ = $$-28121344$$ = $$-1 \cdot 2^{8} \cdot 109849$$ $$J_{10}$$ = $$-409600$$ = $$-1 \cdot 2^{14} \cdot 5^{2}$$ $$g_1$$ = $$62208000$$ $$g_2$$ = $$1702080$$ $$g_3$$ = $$-85248$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2])

## Rational points

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.

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]];

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

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

Number of rational Weierstrass points: $$2$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 12 (p = 2), 1 (p = 5) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{24}\Z$$

### Sato-Tate group

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

### Decomposition

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 $$\GL_2$$-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 $$\Q$$.