# Properties

 Label 563011.a.563011.1 Conductor 563011 Discriminant -563011 Mordell-Weil group $$\Z \times \Z \times \Z \times \Z$$ 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

 $y^2 + (x^3 + x + 1)y = -2x^4 + x^3 + 7x^2 + 4x$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -2x^4z^2 + x^3z^3 + 7x^2z^4 + 4xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 6x^4 + 6x^3 + 29x^2 + 18x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 4, 7, 1, -2], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 4, 7, 1, -2]), R([1, 1, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 18, 29, 6, -6, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$563011$$ = $$563011$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-563011$$ = $$- 563011$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$2760$$ = $$2^{3} \cdot 3 \cdot 5 \cdot 23$$ $$I_4$$ = $$188772$$ = $$2^{2} \cdot 3 \cdot 15731$$ $$I_6$$ = $$145837416$$ = $$2^{3} \cdot 3 \cdot 2287 \cdot 2657$$ $$I_{10}$$ = $$-2306093056$$ = $$- 2^{12} \cdot 563011$$ $$J_2$$ = $$345$$ = $$3 \cdot 5 \cdot 23$$ $$J_4$$ = $$2993$$ = $$41 \cdot 73$$ $$J_6$$ = $$30309$$ = $$3 \cdot 10103$$ $$J_8$$ = $$374639$$ = $$374639$$ $$J_{10}$$ = $$-563011$$ = $$- 563011$$ $$g_1$$ = $$-4887597965625/563011$$ $$g_2$$ = $$-122903429625/563011$$ $$g_3$$ = $$-3607528725/563011$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : 2 : 1)$$ $$(2 : 1 : 1)$$ $$(-2 : 4 : 1)$$ $$(1 : -5 : 1)$$ $$(-2 : 5 : 1)$$ $$(4 : -5 : 1)$$
$$(-3 : 6 : 1)$$ $$(-5 : 9 : 6)$$ $$(2 : -12 : 1)$$ $$(-3 : 15 : 2)$$ $$(-3 : 16 : 2)$$ $$(-3 : 23 : 1)$$
$$(4 : -64 : 1)$$ $$(-7 : 64 : 6)$$ $$(1 : 75 : 5)$$ $$(-5 : 80 : 6)$$ $$(1 : -226 : 5)$$ $$(-7 : 315 : 6)$$
$$(57 : -12768 : 20)$$ $$(57 : -203225 : 20)$$

magma: [C![-7,64,6],C![-7,315,6],C![-5,9,6],C![-5,80,6],C![-3,6,1],C![-3,15,2],C![-3,16,2],C![-3,23,1],C![-2,4,1],C![-2,5,1],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-226,5],C![1,-5,1],C![1,-1,0],C![1,0,0],C![1,2,1],C![1,75,5],C![2,-12,1],C![2,1,1],C![4,-64,1],C![4,-5,1],C![57,-203225,20],C![57,-12768,20]];

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-2 : 5 : 1) - (1 : 0 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 3z^3$$ $$0.821823$$ $$\infty$$
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.304182$$ $$\infty$$
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.786746$$ $$\infty$$
$$(-1 : 0 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.606468$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$4$$   (upper bound) Mordell-Weil rank: $$4$$ 2-Selmer rank: $$4$$ Regulator: $$0.073355$$ Real period: $$16.33052$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$1.197932$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$563011$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 120 T + 563011 T^{2} )$$

## 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 $$\Q$$.