# Properties

 Label 39993.a.119979.1 Conductor 39993 Discriminant 119979 Mordell-Weil group $$\Z \times \Z \times \Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

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

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

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

magma: X,pi:= SimplifiedModel(C);

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

## Invariants

 Conductor: $$N$$ = $$39993$$ = $$3 \cdot 13331$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$119979$$ = $$3^{2} \cdot 13331$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$520$$ = $$2^{3} \cdot 5 \cdot 13$$ $$I_4$$ = $$76132$$ = $$2^{2} \cdot 7 \cdot 2719$$ $$I_6$$ = $$5394536$$ = $$2^{3} \cdot 7 \cdot 96331$$ $$I_{10}$$ = $$491433984$$ = $$2^{12} \cdot 3^{2} \cdot 13331$$ $$J_2$$ = $$65$$ = $$5 \cdot 13$$ $$J_4$$ = $$-617$$ = $$- 617$$ $$J_6$$ = $$5589$$ = $$3^{5} \cdot 23$$ $$J_8$$ = $$-4351$$ = $$- 19 \cdot 229$$ $$J_{10}$$ = $$119979$$ = $$3^{2} \cdot 13331$$ $$g_1$$ = $$1160290625/119979$$ $$g_2$$ = $$-169443625/119979$$ $$g_3$$ = $$2623725/13331$$

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

## 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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(-1 : -1 : 1)$$ $$(-1 : 0 : 2)$$ $$(2 : 2 : 1)$$ $$(1 : -3 : 1)$$ $$(-2 : -3 : 1)$$ $$(-2 : 6 : 1)$$
$$(-1 : -9 : 2)$$ $$(2 : -15 : 1)$$ $$(-3 : -15 : 2)$$ $$(-3 : 16 : 2)$$ $$(1 : -56 : 5)$$ $$(1 : -63 : 6)$$
$$(1 : -75 : 5)$$ $$(1 : -160 : 6)$$

magma: [C![-3,-15,2],C![-3,16,2],C![-2,-3,1],C![-2,6,1],C![-1,-9,2],C![-1,-1,1],C![-1,0,1],C![-1,0,2],C![0,-1,1],C![0,0,1],C![1,-160,6],C![1,-75,5],C![1,-63,6],C![1,-56,5],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-15,1],C![2,2,1]];

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: $$\Z \times \Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : 0 : 1) - (1 : -1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.390167$$ $$\infty$$
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.456379$$ $$\infty$$
$$(-1 : -1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - 2z^3$$ $$0.094506$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.013478$$ Real period: $$18.81218$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$0.507135$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$3$$ $$2$$ $$1$$ $$2$$ $$( 1 + T )( 1 + 2 T + 3 T^{2} )$$
$$13331$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 70 T + 13331 T^{2} )$$

## 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 $$\Q$$.