# Properties

 Label 1042.a.1042.1 Conductor $1042$ Discriminant $1042$ Mordell-Weil group $$\Z/{9}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x)y = -x^4 - x^3 - x^2 + 2x + 2$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = -x^4z^2 - x^3z^3 - x^2z^4 + 2xz^5 + 2z^6$ (dehomogenize, simplify) $y^2 = x^6 - 2x^4 - 4x^3 - 3x^2 + 8x + 8$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([8, 8, -3, -4, -2, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1042$$ $$=$$ $$2 \cdot 521$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$1042$$ $$=$$ $$2 \cdot 521$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$480$$ $$=$$ $$2^{5} \cdot 3 \cdot 5$$ $$I_4$$ $$=$$ $$3912$$ $$=$$ $$2^{3} \cdot 3 \cdot 163$$ $$I_6$$ $$=$$ $$728889$$ $$=$$ $$3 \cdot 7 \cdot 61 \cdot 569$$ $$I_{10}$$ $$=$$ $$-4168$$ $$=$$ $$- 2^{3} \cdot 521$$ $$J_2$$ $$=$$ $$240$$ $$=$$ $$2^{4} \cdot 3 \cdot 5$$ $$J_4$$ $$=$$ $$1748$$ $$=$$ $$2^{2} \cdot 19 \cdot 23$$ $$J_6$$ $$=$$ $$-5521$$ $$=$$ $$-5521$$ $$J_8$$ $$=$$ $$-1095136$$ $$=$$ $$- 2^{5} \cdot 7 \cdot 4889$$ $$J_{10}$$ $$=$$ $$-1042$$ $$=$$ $$- 2 \cdot 521$$ $$g_1$$ $$=$$ $$-398131200000/521$$ $$g_2$$ $$=$$ $$-12082176000/521$$ $$g_3$$ $$=$$ $$159004800/521$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

magma: [C![-1,1,1],C![1,-1,0],C![1,0,0]]; // minimal model

magma: [C![-1,0,1],C![1,-1,0],C![1,1,0]]; // simplified model

Number of rational Weierstrass points: $$1$$

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: $$\Z/{9}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0$$ $$9$$
Generator $D_0$ Height Order
$$(-1 : 1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0$$ $$9$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + xz^2 + 2z^3$$ $$0$$ $$9$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$30.42301$$ Tamagawa product: $$1$$ Torsion order: $$9$$ Leading coefficient: $$0.375592$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T^{2} )$$
$$521$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 27 T + 521 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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