# Properties

 Label 576.b.147456.1 Conductor 576 Discriminant -147456 Sato-Tate group $E_1$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 0, 2, 0, 2, 0, 1], R![]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 0, 2, 0, 2, 0, 1]), R([]))

$y^2 = x^6 + 2x^4 + 2x^2 + 1$

## Invariants

 magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(576,2),R![1]>*])); Factorization($1); $$N$$ = $$576$$ = $$2^{6} \cdot 3^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-147456$$ = $$-1 \cdot 2^{14} \cdot 3^{2}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-4864$$ = $$-1 \cdot 2^{8} \cdot 19$$ $$I_4$$ = $$111616$$ = $$2^{10} \cdot 109$$ $$I_6$$ = $$-179208192$$ = $$-1 \cdot 2^{15} \cdot 3 \cdot 1823$$ $$I_{10}$$ = $$-603979776$$ = $$-1 \cdot 2^{26} \cdot 3^{2}$$ $$J_2$$ = $$-608$$ = $$-1 \cdot 2^{5} \cdot 19$$ $$J_4$$ = $$14240$$ = $$2^{5} \cdot 5 \cdot 89$$ $$J_6$$ = $$-405504$$ = $$-1 \cdot 2^{12} \cdot 3^{2} \cdot 11$$ $$J_8$$ = $$10942208$$ = $$2^{8} \cdot 42743$$ $$J_{10}$$ = $$-147456$$ = $$-1 \cdot 2^{14} \cdot 3^{2}$$ $$g_1$$ = $$5071050752/9$$ $$g_2$$ = $$195344320/9$$ $$g_3$$ = $$1016576$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$D_4$$ (GAP id : [8,3]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$D_4$$ (GAP id : [8,3])

## 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![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0]];

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

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 1.0 Real period: 9.3011194758216110390641169009 Tamagawa numbers: 8 (p = 2), 1 (p = 3) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{4}\Z \times \Z/{4}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_1$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

### Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the square of the elliptic curve:
Elliptic curve 24.a4

### Endomorphisms

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ a non-Eichler order of index $$4$$ in a maximal order of $$\End (J_{}) \otimes \Q$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.