# Properties

 Label 59967.a.539703.1 Conductor 59967 Discriminant -539703 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

 $y^2 + (x^3 + x^2 + 1)y = x^4 + 2x^3 - x^2 - 2x$ (homogenize, simplify) $y^2 + (x^3 + x^2z + z^3)y = x^4z^2 + 2x^3z^3 - x^2z^4 - 2xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 + 5x^4 + 10x^3 - 2x^2 - 8x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, -8, -2, 10, 5, 2, 1]))

## Invariants

 Conductor: $$N$$ = $$59967$$ = $$3^{3} \cdot 2221$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-539703$$ = $$- 3^{5} \cdot 2221$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-120$$ = $$- 2^{3} \cdot 3 \cdot 5$$ $$I_4$$ = $$46116$$ = $$2^{2} \cdot 3^{3} \cdot 7 \cdot 61$$ $$I_6$$ = $$-16924248$$ = $$- 2^{3} \cdot 3^{3} \cdot 11 \cdot 17 \cdot 419$$ $$I_{10}$$ = $$-2210623488$$ = $$- 2^{12} \cdot 3^{5} \cdot 2221$$ $$J_2$$ = $$-15$$ = $$- 3 \cdot 5$$ $$J_4$$ = $$-471$$ = $$- 3 \cdot 157$$ $$J_6$$ = $$27373$$ = $$31 \cdot 883$$ $$J_8$$ = $$-158109$$ = $$- 3 \cdot 7 \cdot 7529$$ $$J_{10}$$ = $$-539703$$ = $$- 3^{5} \cdot 2221$$ $$g_1$$ = $$3125/2221$$ $$g_2$$ = $$-19625/6663$$ $$g_3$$ = $$-684325/59967$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(-1 : -1 : 1)$$ $$(-2 : 0 : 1)$$ $$(1 : -3 : 1)$$ $$(-2 : 3 : 1)$$ $$(-1 : 3 : 2)$$ $$(3 : 3 : 1)$$
$$(3 : 7 : 2)$$ $$(-2 : 9 : 3)$$ $$(-1 : -12 : 2)$$ $$(-7 : -15 : 2)$$ $$(3 : -40 : 1)$$ $$(-2 : -40 : 3)$$
$$(3 : -60 : 2)$$ $$(-7 : 252 : 2)$$

magma: [C![-7,-15,2],C![-7,252,2],C![-2,-40,3],C![-2,0,1],C![-2,3,1],C![-2,9,3],C![-1,-12,2],C![-1,-1,1],C![-1,0,1],C![-1,3,2],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![3,-60,2],C![3,-40,1],C![3,3,1],C![3,7,2]];

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
$$(-2 : 0 : 1) - (1 : -1 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.398690$$ $$\infty$$
$$(-1 : -1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - 2z^3$$ $$0.228205$$ $$\infty$$
$$(-2 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.192414$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.013598$$ Real period: $$14.89874$$ Tamagawa product: $$3$$ Torsion order: $$1$$ Leading coefficient: $$0.607807$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$3$$ $$5$$ $$3$$ $$3$$ $$1 + 3 T + 3 T^{2}$$
$$2221$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 76 T + 2221 T^{2} )$$

## 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 $$\Q$$.