# Properties

 Label 10005.b.450225.1 Conductor $10005$ Discriminant $450225$ Mordell-Weil group $$\Z \times \Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -2x^4 + 3x^2 + x + 2$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -2x^4z^2 + 3x^2z^4 + xz^5 + 2z^6$ (dehomogenize, simplify) $y^2 = x^6 - 6x^4 + 2x^3 + 13x^2 + 6x + 9$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 1, 3, 0, -2]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 1, 3, 0, -2], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([9, 6, 13, 2, -6, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10005$$ $$=$$ $$3 \cdot 5 \cdot 23 \cdot 29$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$450225$$ $$=$$ $$3^{3} \cdot 5^{2} \cdot 23 \cdot 29$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$444$$ $$=$$ $$2^{2} \cdot 3 \cdot 37$$ $$I_4$$ $$=$$ $$108777$$ $$=$$ $$3 \cdot 101 \cdot 359$$ $$I_6$$ $$=$$ $$21372411$$ $$=$$ $$3 \cdot 137 \cdot 149 \cdot 349$$ $$I_{10}$$ $$=$$ $$-57628800$$ $$=$$ $$- 2^{7} \cdot 3^{3} \cdot 5^{2} \cdot 23 \cdot 29$$ $$J_2$$ $$=$$ $$111$$ $$=$$ $$3 \cdot 37$$ $$J_4$$ $$=$$ $$-4019$$ $$=$$ $$-4019$$ $$J_6$$ $$=$$ $$-153925$$ $$=$$ $$- 5^{2} \cdot 47 \cdot 131$$ $$J_8$$ $$=$$ $$-8309509$$ $$=$$ $$- 13 \cdot 23 \cdot 27791$$ $$J_{10}$$ $$=$$ $$-450225$$ $$=$$ $$- 3^{3} \cdot 5^{2} \cdot 23 \cdot 29$$ $$g_1$$ $$=$$ $$-624095613/16675$$ $$g_2$$ $$=$$ $$203574407/16675$$ $$g_3$$ $$=$$ $$8428933/2001$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 1)$$ $$(1 : 1 : 1)$$ $$(0 : -2 : 1)$$
$$(-1 : 2 : 1)$$ $$(1 : -4 : 1)$$ $$(-2 : 4 : 1)$$ $$(-2 : 5 : 1)$$ $$(3 : -5 : 1)$$ $$(-3 : 8 : 2)$$
$$(-3 : 23 : 2)$$ $$(3 : -26 : 1)$$
Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 1)$$ $$(1 : 1 : 1)$$ $$(0 : -2 : 1)$$
$$(-1 : 2 : 1)$$ $$(1 : -4 : 1)$$ $$(-2 : 4 : 1)$$ $$(-2 : 5 : 1)$$ $$(3 : -5 : 1)$$ $$(-3 : 8 : 2)$$
$$(-3 : 23 : 2)$$ $$(3 : -26 : 1)$$
Known points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(-2 : -1 : 1)$$ $$(-2 : 1 : 1)$$ $$(0 : -3 : 1)$$ $$(0 : 3 : 1)$$
$$(-1 : -3 : 1)$$ $$(-1 : 3 : 1)$$ $$(1 : -5 : 1)$$ $$(1 : 5 : 1)$$ $$(-3 : -15 : 2)$$ $$(-3 : 15 : 2)$$
$$(3 : -21 : 1)$$ $$(3 : 21 : 1)$$

magma: [C![-3,8,2],C![-3,23,2],C![-2,4,1],C![-2,5,1],C![-1,-1,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,1,1],C![3,-26,1],C![3,-5,1]]; // minimal model

magma: [C![-3,-15,2],C![-3,15,2],C![-2,-1,1],C![-2,1,1],C![-1,-3,1],C![-1,3,1],C![0,-3,1],C![0,3,1],C![1,-5,1],C![1,-1,0],C![1,1,0],C![1,5,1],C![3,-21,1],C![3,21,1]]; // 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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 + z^3$$ $$0.061340$$ $$\infty$$
$$(-2 : 4 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 + 2z^3$$ $$0.075791$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 + z^3$$ $$0.061340$$ $$\infty$$
$$(-2 : 4 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 + 2z^3$$ $$0.075791$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - xz^2 + 3z^3$$ $$0.061340$$ $$\infty$$
$$(-2 : -1 : 1) + (1 : 5 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(x - z) (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - xz^2 + 5z^3$$ $$0.075791$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.003683$$ Real period: $$16.51737$$ Tamagawa product: $$6$$ Torsion order: $$1$$ Leading coefficient: $$0.365022$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$1$$ $$3$$ $$3$$ $$( 1 - T )( 1 + 3 T + 3 T^{2} )$$
$$5$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + 2 T + 5 T^{2} )$$
$$23$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 6 T + 23 T^{2} )$$
$$29$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 29 T^{2} )$$

## 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 $$\Q$$.