# Properties

 Label 4128.b.594432.1 Conductor 4128 Discriminant -594432 Mordell-Weil group $$\Z$$ 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: SageMath / Magma

## Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([1, 4, 4, -4, 0, 0, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$4128$$ $$=$$ $$2^{5} \cdot 3 \cdot 43$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-594432$$ $$=$$ $$- 2^{9} \cdot 3^{3} \cdot 43$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-864$$ $$=$$ $$- 2^{5} \cdot 3^{3}$$ $$I_4$$ $$=$$ $$105024$$ $$=$$ $$2^{6} \cdot 3 \cdot 547$$ $$I_6$$ $$=$$ $$-25155072$$ $$=$$ $$- 2^{9} \cdot 3^{2} \cdot 53 \cdot 103$$ $$I_{10}$$ $$=$$ $$-2434793472$$ $$=$$ $$- 2^{21} \cdot 3^{3} \cdot 43$$ $$J_2$$ $$=$$ $$-108$$ $$=$$ $$- 2^{2} \cdot 3^{3}$$ $$J_4$$ $$=$$ $$-608$$ $$=$$ $$- 2^{5} \cdot 19$$ $$J_6$$ $$=$$ $$7936$$ $$=$$ $$2^{8} \cdot 31$$ $$J_8$$ $$=$$ $$-306688$$ $$=$$ $$- 2^{9} \cdot 599$$ $$J_{10}$$ $$=$$ $$-594432$$ $$=$$ $$- 2^{9} \cdot 3^{3} \cdot 43$$ $$g_1$$ $$=$$ $$1062882/43$$ $$g_2$$ $$=$$ $$-55404/43$$ $$g_3$$ $$=$$ $$-6696/43$$

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
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 1 : 1)$$ $$(1 : 1 : 1)$$
$$(-1 : -2 : 1)$$ $$(-1 : -1 : 2)$$ $$(1 : -2 : 1)$$ $$(-1 : -7 : 2)$$

magma: [C![-1,-7,2],C![-1,-2,1],C![-1,-1,2],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,1,1]];

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -7 : 2) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(x - z) (2x + z)$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$5xz^2 - z^3$$ $$0.001730$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.001730$$ Real period: $$13.79653$$ Tamagawa product: $$24$$ Torsion order: $$1$$ Leading coefficient: $$0.572871$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$5$$ $$9$$ $$8$$ $$1$$
$$3$$ $$1$$ $$3$$ $$3$$ $$( 1 - T )( 1 + 3 T + 3 T^{2} )$$
$$43$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 43 T^{2} )$$

## 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 $$\Q$$.