# Properties

 Label 8649.a.233523.1 Conductor 8649 Discriminant 233523 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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-70, -47, 7, 3, -3], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-70, -47, 7, 3, -3]), R([1, 1, 0, 1]))

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-70, -47, 7, 3, -3], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-279, -186, 29, 14, -10, 0, 1]))

 $y^2 + (x^3 + x + 1)y = -3x^4 + 3x^3 + 7x^2 - 47x - 70$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -3x^4z^2 + 3x^3z^3 + 7x^2z^4 - 47xz^5 - 70z^6$ (dehomogenize, simplify) $y^2 = x^6 - 10x^4 + 14x^3 + 29x^2 - 186x - 279$ (minimize, homogenize)

## Invariants

 $$N$$ = $$8649$$ = $$3^{2} \cdot 31^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$233523$$ = $$3^{5} \cdot 31^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$72776$$ = $$2^{3} \cdot 11 \cdot 827$$ $$I_4$$ = $$-4565084$$ = $$- 2^{2} \cdot 1141271$$ $$I_6$$ = $$-110566728408$$ = $$- 2^{3} \cdot 3 \cdot 4606947017$$ $$I_{10}$$ = $$956510208$$ = $$2^{12} \cdot 3^{5} \cdot 31^{2}$$ $$J_2$$ = $$9097$$ = $$11 \cdot 827$$ $$J_4$$ = $$3495695$$ = $$5 \cdot 7 \cdot 99877$$ $$J_6$$ = $$1814445117$$ = $$3^{4} \cdot 29 \cdot 107 \cdot 7219$$ $$J_8$$ = $$1071530924081$$ = $$23 \cdot 157 \cdot 296740771$$ $$J_{10}$$ = $$233523$$ = $$3^{5} \cdot 31^{2}$$ $$g_1$$ = $$62300419867534985257/233523$$ $$g_2$$ = $$2631649929116327735/233523$$ $$g_3$$ = $$1853767256362813/2883$$

## Automorphism group

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

## Rational points

magma: [C![-7,189,3],C![-7,190,3],C![-3,10,1],C![-3,19,1],C![-2,4,1],C![-2,5,1],C![1,-1,0],C![1,0,0]];

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-2 : 4 : 1)$$ $$(-2 : 5 : 1)$$ $$(-3 : 10 : 1)$$ $$(-3 : 19 : 1)$$
$$(-7 : 189 : 3)$$ $$(-7 : 190 : 3)$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$0$$

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.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z \times \Z$$

Generator Height Order
$$z (3x + 7z)$$ $$=$$ $$0,$$ $$3y$$ $$=$$ $$-3x^3 - 17z^3$$ $$0.132768$$ $$\infty$$
$$z (x + 3z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$10z^3$$ $$0.141745$$ $$\infty$$

## BSD invariants

 Analytic rank: $$2$$   (upper bound) Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.018738$$ Real period: $$9.071150$$ Tamagawa product: $$2$$ 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
$$3$$ $$5$$ $$2$$ $$2$$ $$( 1 + T )^{2}$$
$$31$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$

## 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 $$\Q$$:

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

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