# Properties

 Label 1109.c.1109.1 Conductor $1109$ Discriminant $1109$ Mordell-Weil group $$\Z/{5}\Z$$ 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

 $y^2 + (x^3 + x)y = x^5 - 2x^3 - 2x^2 - 1$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = x^5z - 2x^3z^3 - 2x^2z^4 - z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 + 2x^4 - 8x^3 - 7x^2 - 4$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([-4, 0, -7, -8, 2, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1109$$ $$=$$ $$1109$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$1109$$ $$=$$ $$1109$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$392$$ $$=$$ $$2^{3} \cdot 7^{2}$$ $$I_4$$ $$=$$ $$292$$ $$=$$ $$2^{2} \cdot 73$$ $$I_6$$ $$=$$ $$36703$$ $$=$$ $$17^{2} \cdot 127$$ $$I_{10}$$ $$=$$ $$4436$$ $$=$$ $$2^{2} \cdot 1109$$ $$J_2$$ $$=$$ $$196$$ $$=$$ $$2^{2} \cdot 7^{2}$$ $$J_4$$ $$=$$ $$1552$$ $$=$$ $$2^{4} \cdot 97$$ $$J_6$$ $$=$$ $$16001$$ $$=$$ $$16001$$ $$J_8$$ $$=$$ $$181873$$ $$=$$ $$181873$$ $$J_{10}$$ $$=$$ $$1109$$ $$=$$ $$1109$$ $$g_1$$ $$=$$ $$289254654976/1109$$ $$g_2$$ $$=$$ $$11685839872/1109$$ $$g_3$$ $$=$$ $$614694416/1109$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1)$$ All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1)$$ All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : 0 : 1)$$

magma: [C![-2,5,1],C![1,-1,0],C![1,0,0]]; // minimal model

magma: [C![-2,0,1],C![1,-1,0],C![1,1,0]]; // 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: $$\Z/{5}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-2 : 5 : 1) - (1 : -1 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$5z^3$$ $$0$$ $$5$$
Generator $D_0$ Height Order
$$(-2 : 5 : 1) - (1 : -1 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$5z^3$$ $$0$$ $$5$$
Generator $D_0$ Height Order
$$(-2 : 0 : 1) - (1 : -1 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + xz^2 + 10z^3$$ $$0$$ $$5$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$9.552149$$ Tamagawa product: $$1$$ Torsion order: $$5$$ Leading coefficient: $$0.382085$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$1109$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 60 T + 1109 T^{2} )$$

## 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 $$\Q$$.