# Properties

 Label 576.a.576.1 Conductor 576 Discriminant -576 Sato-Tate group $E_2$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

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

$y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$576$$ = $$2^{6} \cdot 3^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-576$$ = $$-1 \cdot 2^{6} \cdot 3^{2}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-544$$ = $$-1 \cdot 2^{5} \cdot 17$$ $$I_4$$ = $$7936$$ = $$2^{8} \cdot 31$$ $$I_6$$ = $$-1339392$$ = $$-1 \cdot 2^{12} \cdot 3 \cdot 109$$ $$I_{10}$$ = $$-2359296$$ = $$-1 \cdot 2^{18} \cdot 3^{2}$$ $$J_2$$ = $$-68$$ = $$-1 \cdot 2^{2} \cdot 17$$ $$J_4$$ = $$110$$ = $$2 \cdot 5 \cdot 11$$ $$J_6$$ = $$36$$ = $$2^{2} \cdot 3^{2}$$ $$J_8$$ = $$-3637$$ = $$-1 \cdot 3637$$ $$J_{10}$$ = $$-576$$ = $$-1 \cdot 2^{6} \cdot 3^{2}$$ $$g_1$$ = $$22717712/9$$ $$g_2$$ = $$540430/9$$ $$g_3$$ = $$-289$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_4$$ (GAP id : [4,1]) 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![0,0,1],C![1,-1,0],C![1,0,0]];

All rational points: (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : 0 : 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: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 2), 1 (p = 3) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{10}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_2$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

### Decomposition

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\sqrt{2})$$ with defining polynomial:
$$x^{2} - 2$$

Decomposes up to isogeny as the square of the elliptic curve:
Elliptic curve 2.2.8.1-9.1-a3

### Endomorphisms

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

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

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

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 $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$