# Properties

 Label 10000.a.160000.1 Conductor $10000$ Discriminant $-160000$ Mordell-Weil group $$\Z/{5}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

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Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + xy = x^6 - 2x^5 + 2x^4 - x^3 + 2x^2 - 3x + 2$ (homogenize, simplify) $y^2 + xz^2y = x^6 - 2x^5z + 2x^4z^2 - x^3z^3 + 2x^2z^4 - 3xz^5 + 2z^6$ (dehomogenize, simplify) $y^2 = 4x^6 - 8x^5 + 8x^4 - 4x^3 + 9x^2 - 12x + 8$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -3, 2, -1, 2, -2, 1]), R([0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -3, 2, -1, 2, -2, 1], R![0, 1]);

sage: X = HyperellipticCurve(R([8, -12, 9, -4, 8, -8, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10000$$ $$=$$ $$2^{4} \cdot 5^{4}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-160000$$ $$=$$ $$- 2^{8} \cdot 5^{4}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$612$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 17$$ $$I_4$$ $$=$$ $$11325$$ $$=$$ $$3 \cdot 5^{2} \cdot 151$$ $$I_6$$ $$=$$ $$1916325$$ $$=$$ $$3^{3} \cdot 5^{2} \cdot 17 \cdot 167$$ $$I_{10}$$ $$=$$ $$20000$$ $$=$$ $$2^{5} \cdot 5^{4}$$ $$J_2$$ $$=$$ $$612$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 17$$ $$J_4$$ $$=$$ $$8056$$ $$=$$ $$2^{3} \cdot 19 \cdot 53$$ $$J_6$$ $$=$$ $$110704$$ $$=$$ $$2^{4} \cdot 11 \cdot 17 \cdot 37$$ $$J_8$$ $$=$$ $$712928$$ $$=$$ $$2^{5} \cdot 22279$$ $$J_{10}$$ $$=$$ $$160000$$ $$=$$ $$2^{8} \cdot 5^{4}$$ $$g_1$$ $$=$$ $$335364543972/625$$ $$g_2$$ $$=$$ $$7213296078/625$$ $$g_3$$ $$=$$ $$161966871/625$$

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

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

magma: [C![1,-2,0],C![1,2,0]]; // 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: $$\Z/{5}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$8.650087$$ Tamagawa product: $$5$$ Torsion order: $$5$$ Leading coefficient: $$1.730017$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$8$$ $$5$$ $$1 - T$$
$$5$$ $$4$$ $$4$$ $$1$$ $$1$$

## Sato-Tate group

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

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 50.b
Elliptic curve isogeny class 200.e

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$4$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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