# Properties

 Label 10081.a.10081.1 Conductor $10081$ Discriminant $-10081$ Mordell-Weil group $$\Z \times \Z/{3}\Z$$ 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

 $y^2 + (x^2 + x + 1)y = x^6 + 2x^4 + x^3 + x^2$ (homogenize, simplify) $y^2 + (x^2z + xz^2 + z^3)y = x^6 + 2x^4z^2 + x^3z^3 + x^2z^4$ (dehomogenize, simplify) $y^2 = 4x^6 + 9x^4 + 6x^3 + 7x^2 + 2x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, 2, 7, 6, 9, 0, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10081$$ $$=$$ $$17 \cdot 593$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-10081$$ $$=$$ $$- 17 \cdot 593$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$876$$ $$=$$ $$2^{2} \cdot 3 \cdot 73$$ $$I_4$$ $$=$$ $$8337$$ $$=$$ $$3 \cdot 7 \cdot 397$$ $$I_6$$ $$=$$ $$1989399$$ $$=$$ $$3 \cdot 751 \cdot 883$$ $$I_{10}$$ $$=$$ $$1290368$$ $$=$$ $$2^{7} \cdot 17 \cdot 593$$ $$J_2$$ $$=$$ $$219$$ $$=$$ $$3 \cdot 73$$ $$J_4$$ $$=$$ $$1651$$ $$=$$ $$13 \cdot 127$$ $$J_6$$ $$=$$ $$17815$$ $$=$$ $$5 \cdot 7 \cdot 509$$ $$J_8$$ $$=$$ $$293921$$ $$=$$ $$251 \cdot 1171$$ $$J_{10}$$ $$=$$ $$10081$$ $$=$$ $$17 \cdot 593$$ $$g_1$$ $$=$$ $$503756397099/10081$$ $$g_2$$ $$=$$ $$17341210809/10081$$ $$g_3$$ $$=$$ $$854425215/10081$$

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

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

magma: [C![0,-1,1],C![0,1,1],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: $$\Z \times \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.569377$$ $$\infty$$
$$2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$3$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.569377$$ $$\infty$$
$$2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$3$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 2 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2x^3 + x^2z + xz^2 - z^3$$ $$0.569377$$ $$\infty$$
$$2 \cdot(0 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z + xz^2 + z^3$$ $$0$$ $$3$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.569377$$ Real period: $$13.17825$$ Tamagawa product: $$1$$ Torsion order: $$3$$ Leading coefficient: $$0.833712$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$17$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 3 T + 17 T^{2} )$$
$$593$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 39 T + 593 T^{2} )$$

## 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 $$\Q$$.