# Properties

 Label 1152.a.147456.1 Conductor 1152 Discriminant 147456 Sato-Tate group $J(E_1)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# 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(1152,2),R![1]>*])); Factorization($1); $$N$$ = $$1152$$ = $$2^{7} \cdot 3^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$147456$$ = $$2^{14} \cdot 3^{2}$$

### G2 invariants

magma: G2Invariants(C);

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

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2]) 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![-1,0,1],C![1,-1,0],C![1,0,1],C![1,1,0]];

All rational points: (-1 : 0 : 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: 4 (p = 2), 1 (p = 3) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{8}\Z$$

### Sato-Tate group

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

### Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 24.a5
Elliptic curve 48.a5

### Endomorphisms

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-1})$$ with defining polynomial $$x^{2} + 1$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

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