# Properties

 Label 10086.a.181548.1 Conductor $10086$ Discriminant $181548$ Mordell-Weil group $$\Z \times \Z/{2}\Z$$ 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

 $y^2 + (x^2 + x)y = x^5 + x^4 + x^3 + 2x^2 + 2x + 1$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^5z + x^4z^2 + x^3z^3 + 2x^2z^4 + 2xz^5 + z^6$ (dehomogenize, simplify) $y^2 = 4x^5 + 5x^4 + 6x^3 + 9x^2 + 8x + 4$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([4, 8, 9, 6, 5, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10086$$ $$=$$ $$2 \cdot 3 \cdot 41^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$181548$$ $$=$$ $$2^{2} \cdot 3^{3} \cdot 41^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$388$$ $$=$$ $$2^{2} \cdot 97$$ $$I_4$$ $$=$$ $$11521$$ $$=$$ $$41 \cdot 281$$ $$I_6$$ $$=$$ $$1128033$$ $$=$$ $$3^{3} \cdot 41 \cdot 1019$$ $$I_{10}$$ $$=$$ $$23238144$$ $$=$$ $$2^{9} \cdot 3^{3} \cdot 41^{2}$$ $$J_2$$ $$=$$ $$97$$ $$=$$ $$97$$ $$J_4$$ $$=$$ $$-88$$ $$=$$ $$- 2^{3} \cdot 11$$ $$J_6$$ $$=$$ $$-620$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 31$$ $$J_8$$ $$=$$ $$-16971$$ $$=$$ $$- 3 \cdot 5657$$ $$J_{10}$$ $$=$$ $$181548$$ $$=$$ $$2^{2} \cdot 3^{3} \cdot 41^{2}$$ $$g_1$$ $$=$$ $$8587340257/181548$$ $$g_2$$ $$=$$ $$-20078806/45387$$ $$g_3$$ $$=$$ $$-1458395/45387$$

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: $$(1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 2 : 1),\, (1 : -4 : 1)$$ All points: $$(1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 2 : 1),\, (1 : -4 : 1)$$ All points: $$(1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -2 : 1),\, (0 : 2 : 1),\, (1 : -6 : 1),\, (1 : 6 : 1)$$

magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-4,1],C![1,0,0],C![1,2,1]]; // minimal model

magma: [C![-1,0,1],C![0,-2,1],C![0,2,1],C![1,-6,1],C![1,0,0],C![1,6,1]]; // simplified model

Number of rational Weierstrass points: $$2$$

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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) + (1 : -4 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2 - z^3$$ $$0.058560$$ $$\infty$$
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) + (1 : -4 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2 - z^3$$ $$0.058560$$ $$\infty$$
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(0 : -2 : 1) + (1 : -6 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - 5xz^2 - 2z^3$$ $$0.058560$$ $$\infty$$
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z + xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.058560$$ Real period: $$10.58502$$ Tamagawa product: $$6$$ Torsion order: $$2$$ Leading coefficient: $$0.929797$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 - T + 2 T^{2} )$$
$$3$$ $$1$$ $$3$$ $$3$$ $$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$41$$ $$2$$ $$2$$ $$1$$ $$1 + 2 T + 41 T^{2}$$

## 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 $$\Q$$.