# Properties

 Label 1094.a.2188.1 Conductor $1094$ Discriminant $2188$ Mordell-Weil group $$\Z$$ 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

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

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

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

sage: X = HyperellipticCurve(R([1, 0, -4, 2, 4, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1094$$ $$=$$ $$2 \cdot 547$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$2188$$ $$=$$ $$2^{2} \cdot 547$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$20$$ $$=$$ $$2^{2} \cdot 5$$ $$I_4$$ $$=$$ $$3001$$ $$=$$ $$3001$$ $$I_6$$ $$=$$ $$-30387$$ $$=$$ $$- 3 \cdot 7 \cdot 1447$$ $$I_{10}$$ $$=$$ $$280064$$ $$=$$ $$2^{9} \cdot 547$$ $$J_2$$ $$=$$ $$5$$ $$=$$ $$5$$ $$J_4$$ $$=$$ $$-124$$ $$=$$ $$- 2^{2} \cdot 31$$ $$J_6$$ $$=$$ $$596$$ $$=$$ $$2^{2} \cdot 149$$ $$J_8$$ $$=$$ $$-3099$$ $$=$$ $$- 3 \cdot 1033$$ $$J_{10}$$ $$=$$ $$2188$$ $$=$$ $$2^{2} \cdot 547$$ $$g_1$$ $$=$$ $$3125/2188$$ $$g_2$$ $$=$$ $$-3875/547$$ $$g_3$$ $$=$$ $$3725/547$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(1 : -2 : 1)$$ $$(-1 : -3 : 2)$$ $$(-1 : -4 : 2)$$
All points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(1 : -2 : 1)$$ $$(-1 : -3 : 2)$$ $$(-1 : -4 : 2)$$
All points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 2)$$
$$(-1 : 1 : 2)$$ $$(1 : -2 : 1)$$ $$(1 : 2 : 1)$$

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

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

Number of rational Weierstrass points: $$1$$

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
$$(-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0.003585$$ $$\infty$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0.003585$$ $$\infty$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 2xz^2 - z^3$$ $$0.003585$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.003585$$ Real period: $$26.60554$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$0.190768$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + T + 2 T^{2} )$$
$$547$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 19 T + 547 T^{2} )$$

## 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 $$\Q$$.