# Properties

 Label 440509.a.440509.1 Conductor $440509$ Discriminant $440509$ Mordell-Weil group $$\Z \times \Z \times \Z \times \Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x + 1)y = x^5 - x^4 - 5x^3 + 9x + 6$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = x^5z - x^4z^2 - 5x^3z^3 + 9xz^5 + 6z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 2x^4 - 18x^3 + x^2 + 38x + 25$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([6, 9, 0, -5, -1, 1]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![6, 9, 0, -5, -1, 1], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([25, 38, 1, -18, -2, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$440509$$ $$=$$ $$440509$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$440509$$ $$=$$ $$440509$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1028$$ $$=$$ $$2^{2} \cdot 257$$ $$I_4$$ $$=$$ $$71593$$ $$=$$ $$71593$$ $$I_6$$ $$=$$ $$17110373$$ $$=$$ $$7 \cdot 79 \cdot 30941$$ $$I_{10}$$ $$=$$ $$56385152$$ $$=$$ $$2^{7} \cdot 440509$$ $$J_2$$ $$=$$ $$257$$ $$=$$ $$257$$ $$J_4$$ $$=$$ $$-231$$ $$=$$ $$- 3 \cdot 7 \cdot 11$$ $$J_6$$ $$=$$ $$14605$$ $$=$$ $$5 \cdot 23 \cdot 127$$ $$J_8$$ $$=$$ $$925031$$ $$=$$ $$173 \cdot 5347$$ $$J_{10}$$ $$=$$ $$440509$$ $$=$$ $$440509$$ $$g_1$$ $$=$$ $$1121154893057/440509$$ $$g_2$$ $$=$$ $$-3921130983/440509$$ $$g_3$$ $$=$$ $$964645645/440509$$

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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-1 : 0 : 1)$$ $$(-1 : 1 : 1)$$ $$(0 : 2 : 1)$$ $$(2 : 0 : 1)$$
$$(1 : 2 : 1)$$ $$(0 : -3 : 1)$$ $$(-2 : 4 : 1)$$ $$(1 : -5 : 1)$$ $$(-2 : 5 : 1)$$ $$(7 : 5 : 3)$$
$$(-1 : 10 : 2)$$ $$(-3 : 10 : 2)$$ $$(2 : -11 : 1)$$ $$(-1 : -13 : 2)$$ $$(-3 : 14 : 1)$$ $$(-3 : 15 : 1)$$
$$(-3 : 21 : 2)$$ $$(-4 : 22 : 1)$$ $$(-4 : 45 : 1)$$ $$(-7 : 94 : 6)$$ $$(-7 : 285 : 6)$$ $$(7 : -438 : 3)$$

magma: [C![-7,94,6],C![-7,285,6],C![-4,22,1],C![-4,45,1],C![-3,10,2],C![-3,14,1],C![-3,15,1],C![-3,21,2],C![-2,4,1],C![-2,5,1],C![-1,-13,2],C![-1,0,1],C![-1,1,1],C![-1,10,2],C![0,-3,1],C![0,2,1],C![1,-5,1],C![1,-1,0],C![1,0,0],C![1,2,1],C![2,-11,1],C![2,0,1],C![7,-438,3],C![7,5,3]];

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
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$1.072693$$ $$\infty$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z$$ $$0.404197$$ $$\infty$$
$$(-2 : 4 : 1) - (1 : 0 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 4z^3$$ $$0.548892$$ $$\infty$$
$$(-1 : 0 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.333349$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$4$$   (upper bound) Mordell-Weil rank: $$4$$ 2-Selmer rank: $$4$$ Regulator: $$0.062575$$ Real period: $$17.94863$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$1.123153$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$440509$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 372 T + 440509 T^{2} )$$

## 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 $$\Q$$.

The conductor $440509$ of the Jacobian of this curve is the smallest known for a genus $2$ curve with analytic rank $4$.