# Properties

 Label 10056.a.181008.1 Conductor $10056$ Discriminant $181008$ Mordell-Weil group $$\Z \times \Z/{2}\Z$$ 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

 $y^2 + xy = 2x^5 - 3x^4 + 4x^3 - 2x^2 + x$ (homogenize, simplify) $y^2 + xz^2y = 2x^5z - 3x^4z^2 + 4x^3z^3 - 2x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 8x^5 - 12x^4 + 16x^3 - 7x^2 + 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -2, 4, -3, 2]), R([0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -2, 4, -3, 2], R![0, 1]);

sage: X = HyperellipticCurve(R([0, 4, -7, 16, -12, 8]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10056$$ $$=$$ $$2^{3} \cdot 3 \cdot 419$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$181008$$ $$=$$ $$2^{4} \cdot 3^{3} \cdot 419$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$368$$ $$=$$ $$2^{4} \cdot 23$$ $$I_4$$ $$=$$ $$4132$$ $$=$$ $$2^{2} \cdot 1033$$ $$I_6$$ $$=$$ $$532068$$ $$=$$ $$2^{2} \cdot 3 \cdot 101 \cdot 439$$ $$I_{10}$$ $$=$$ $$724032$$ $$=$$ $$2^{6} \cdot 3^{3} \cdot 419$$ $$J_2$$ $$=$$ $$184$$ $$=$$ $$2^{3} \cdot 23$$ $$J_4$$ $$=$$ $$722$$ $$=$$ $$2 \cdot 19^{2}$$ $$J_6$$ $$=$$ $$-9500$$ $$=$$ $$- 2^{2} \cdot 5^{3} \cdot 19$$ $$J_8$$ $$=$$ $$-567321$$ $$=$$ $$- 3 \cdot 19 \cdot 37 \cdot 269$$ $$J_{10}$$ $$=$$ $$181008$$ $$=$$ $$2^{4} \cdot 3^{3} \cdot 419$$ $$g_1$$ $$=$$ $$13181630464/11313$$ $$g_2$$ $$=$$ $$281106368/11313$$ $$g_3$$ $$=$$ $$-20102000/11313$$

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

 All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 1 : 1),\, (1 : -2 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 1 : 1),\, (1 : -2 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -3 : 1),\, (1 : 3 : 1)$$

magma: [C![0,0,1],C![1,-2,1],C![1,0,0],C![1,1,1]]; // minimal model

magma: [C![0,0,1],C![1,-3,1],C![1,0,0],C![1,3,1]]; // simplified model

Number of rational Weierstrass points: $$2$$

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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.055780$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.055780$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(1 : -3 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - 4z^3$$ $$0.055780$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.055780$$ Real period: $$8.723347$$ Tamagawa product: $$6$$ Torsion order: $$2$$ Leading coefficient: $$0.729895$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$3$$ $$4$$ $$2$$ $$1 + T + 2 T^{2}$$
$$3$$ $$1$$ $$3$$ $$3$$ $$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$419$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 20 T + 419 T^{2} )$$

## 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 $$\Q$$.