# Properties

 Label 576.a.576.1 Conductor 576 Discriminant -576 Mordell-Weil group $$\Z/{10}\Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x$ (homogenize, simplify) $y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^3z^3 - xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 + 3x^4 + 3x^2 - 2x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, -2, 3, 0, 3, 2, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

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

### G2 invariants

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

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_4$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$D_4$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

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

Number of rational Weierstrass points: $$0$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$22.39625$$ Tamagawa product: $$1$$ Torsion order: $$10$$ Leading coefficient: $$0.223962$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$6$$ $$6$$ $$1$$ $$1 + 2 T + 2 T^{2}$$
$$3$$ $$2$$ $$2$$ $$1$$ $$1 + T^{2}$$

## Sato-Tate group

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

## Decomposition of the Jacobian

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 the Jacobian

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