# Properties

 Label 8649.a.77841.1 Conductor 8649 Discriminant 77841 Mordell-Weil group $$\Z \times \Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

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

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, -6, 5, 6, 2, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$8649$$ = $$3^{2} \cdot 31^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$77841$$ = $$3^{4} \cdot 31^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-184$$ = $$- 2^{3} \cdot 23$$ $$I_4$$ = $$70756$$ = $$2^{2} \cdot 7^{2} \cdot 19^{2}$$ $$I_6$$ = $$-4828056$$ = $$- 2^{3} \cdot 3 \cdot 37 \cdot 5437$$ $$I_{10}$$ = $$318836736$$ = $$2^{12} \cdot 3^{4} \cdot 31^{2}$$ $$J_2$$ = $$-23$$ = $$- 23$$ $$J_4$$ = $$-715$$ = $$- 5 \cdot 11 \cdot 13$$ $$J_6$$ = $$3645$$ = $$3^{6} \cdot 5$$ $$J_8$$ = $$-148765$$ = $$- 5 \cdot 29753$$ $$J_{10}$$ = $$77841$$ = $$3^{4} \cdot 31^{2}$$ $$g_1$$ = $$-6436343/77841$$ $$g_2$$ = $$8699405/77841$$ $$g_3$$ = $$23805/961$$

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

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

## 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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : -1 : 1)$$
$$(-2 : 0 : 1)$$ $$(-1 : 2 : 1)$$ $$(1 : -3 : 1)$$ $$(1 : -5 : 2)$$ $$(1 : -8 : 2)$$ $$(4 : 8 : 3)$$
$$(-2 : 9 : 1)$$ $$(4 : -135 : 3)$$

magma: [C![-2,0,1],C![-2,9,1],C![-1,-1,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-8,2],C![1,-5,2],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![4,-135,3],C![4,8,3]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.066384$$ $$\infty$$
$$(-1 : 2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 + z^3$$ $$0.070872$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.004684$$ Real period: $$18.14230$$ Tamagawa product: $$4$$ Torsion order: $$1$$ Leading coefficient: $$0.339964$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$3$$ $$4$$ $$2$$ $$4$$ $$( 1 + T )^{2}$$
$$31$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

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

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{5}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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