# Properties

 Label 2556.a.30672.1 Conductor 2556 Discriminant -30672 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

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$2556$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 71$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-30672$$ $$=$$ $$- 2^{4} \cdot 3^{3} \cdot 71$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-64$$ $$=$$ $$- 2^{6}$$ $$I_4$$ $$=$$ $$23872$$ $$=$$ $$2^{6} \cdot 373$$ $$I_6$$ $$=$$ $$-4051456$$ $$=$$ $$- 2^{9} \cdot 41 \cdot 193$$ $$I_{10}$$ $$=$$ $$-125632512$$ $$=$$ $$- 2^{16} \cdot 3^{3} \cdot 71$$ $$J_2$$ $$=$$ $$-8$$ $$=$$ $$- 2^{3}$$ $$J_4$$ $$=$$ $$-246$$ $$=$$ $$- 2 \cdot 3 \cdot 41$$ $$J_6$$ $$=$$ $$6480$$ $$=$$ $$2^{4} \cdot 3^{4} \cdot 5$$ $$J_8$$ $$=$$ $$-28089$$ $$=$$ $$- 3^{2} \cdot 3121$$ $$J_{10}$$ $$=$$ $$-30672$$ $$=$$ $$- 2^{4} \cdot 3^{3} \cdot 71$$ $$g_1$$ $$=$$ $$2048/1917$$ $$g_2$$ $$=$$ $$-2624/639$$ $$g_3$$ $$=$$ $$-960/71$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : 0 : 1)$$
$$(-1 : 1 : 1)$$ $$(1 : -1 : 1)$$ $$(-2 : 3 : 1)$$ $$(-2 : 5 : 1)$$

magma: [C![-2,3,1],C![-2,5,1],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1]];

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 1 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0.015775$$ $$\infty$$
$$2 \cdot(0 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 + z^3$$ $$0$$ $$3$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.015775$$ Real period: $$20.51510$$ Tamagawa product: $$9$$ Torsion order: $$3$$ Leading coefficient: $$0.323646$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$4$$ $$3$$ $$1 + 2 T + 2 T^{2}$$
$$3$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$
$$71$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 3 T + 71 T^{2} )$$

## 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 $$\Q$$.