# Properties

 Label 5580.a.33480.1 Conductor 5580 Discriminant 33480 Mordell-Weil group $$\Z \times \Z/{3}\Z$$ 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

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

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

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

sage: X = HyperellipticCurve(R([1, 2, 5, -6, -6, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$5580$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 5 \cdot 31$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$33480$$ $$=$$ $$2^{3} \cdot 3^{3} \cdot 5 \cdot 31$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$776$$ $$=$$ $$2^{3} \cdot 97$$ $$I_4$$ $$=$$ $$41668$$ $$=$$ $$2^{2} \cdot 11 \cdot 947$$ $$I_6$$ $$=$$ $$3370568$$ $$=$$ $$2^{3} \cdot 31 \cdot 13591$$ $$I_{10}$$ $$=$$ $$137134080$$ $$=$$ $$2^{15} \cdot 3^{3} \cdot 5 \cdot 31$$ $$J_2$$ $$=$$ $$97$$ $$=$$ $$97$$ $$J_4$$ $$=$$ $$-42$$ $$=$$ $$- 2 \cdot 3 \cdot 7$$ $$J_6$$ $$=$$ $$7956$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 13 \cdot 17$$ $$J_8$$ $$=$$ $$192492$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 5347$$ $$J_{10}$$ $$=$$ $$33480$$ $$=$$ $$2^{3} \cdot 3^{3} \cdot 5 \cdot 31$$ $$g_1$$ $$=$$ $$8587340257/33480$$ $$g_2$$ $$=$$ $$-6388711/5580$$ $$g_3$$ $$=$$ $$2079389/930$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : -1 : 1)$$ $$(1 : -2 : 1)$$ $$(12 : -531 : 7)$$ $$(12 : -2128 : 7)$$

magma: [C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![12,-2128,7],C![12,-531,7]];

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: $$\Z \times \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.027041$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$3$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.027041$$ Real period: $$20.90560$$ Tamagawa product: $$9$$ Torsion order: $$3$$ Leading coefficient: $$0.565311$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$
$$3$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 5 T^{2} )$$
$$31$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + T + 31 T^{2} )$$

## 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 $$\Q$$.