# Properties

 Label 10086.b.413526.1 Conductor $10086$ Discriminant $413526$ Mordell-Weil group $$\Z/{6}\Z$$ 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 for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x)y = -x^6 - 4x^5 - 7x^4 - 6x^3 + 3x + 3$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = -x^6 - 4x^5z - 7x^4z^2 - 6x^3z^3 + 3xz^5 + 3z^6$ (dehomogenize, simplify) $y^2 = -3x^6 - 16x^5 - 26x^4 - 24x^3 + x^2 + 12x + 12$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 3, 0, -6, -7, -4, -1]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 3, 0, -6, -7, -4, -1], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([12, 12, 1, -24, -26, -16, -3]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10086$$ $$=$$ $$2 \cdot 3 \cdot 41^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$413526$$ $$=$$ $$2 \cdot 3 \cdot 41^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1208$$ $$=$$ $$2^{3} \cdot 151$$ $$I_4$$ $$=$$ $$136120$$ $$=$$ $$2^{3} \cdot 5 \cdot 41 \cdot 83$$ $$I_6$$ $$=$$ $$71419827$$ $$=$$ $$3 \cdot 41 \cdot 101 \cdot 5749$$ $$I_{10}$$ $$=$$ $$1654104$$ $$=$$ $$2^{3} \cdot 3 \cdot 41^{3}$$ $$J_2$$ $$=$$ $$604$$ $$=$$ $$2^{2} \cdot 151$$ $$J_4$$ $$=$$ $$-7486$$ $$=$$ $$- 2 \cdot 19 \cdot 197$$ $$J_6$$ $$=$$ $$-3619151$$ $$=$$ $$-3619151$$ $$J_8$$ $$=$$ $$-560501850$$ $$=$$ $$- 2 \cdot 3 \cdot 5^{2} \cdot 29 \cdot 269 \cdot 479$$ $$J_{10}$$ $$=$$ $$413526$$ $$=$$ $$2 \cdot 3 \cdot 41^{3}$$ $$g_1$$ $$=$$ $$40193395584512/206763$$ $$g_2$$ $$=$$ $$-824765797952/206763$$ $$g_3$$ $$=$$ $$-660162095608/206763$$

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

 This curve has no rational points. This curve has no rational points. This curve has no rational points.

magma: []; // minimal model

magma: []; // simplified model

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: $$\Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - 2z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - 2z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 3xz^2 - 4z^3$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$3$$ Regulator: $$1$$ Real period: $$5.766529$$ Tamagawa product: $$2$$ Torsion order: $$6$$ Leading coefficient: $$1.281450$$ Analytic order of Ш: $$4$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T^{2} )$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$41$$ $$2$$ $$3$$ $$2$$ $$1 - 6 T + 41 T^{2}$$

## 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 $$\Q$$.