# Properties

 Label 800.a.8000.1 Conductor $800$ Discriminant $8000$ Mordell-Weil group $$\Z/{4}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x^2 + x + 1)y = -x^6 + 2x^4 + 4x^3 + 2x^2 - 1$ (homogenize, simplify) $y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^6 + 2x^4z^2 + 4x^3z^3 + 2x^2z^4 - z^6$ (dehomogenize, simplify) $y^2 = -3x^6 + 2x^5 + 11x^4 + 20x^3 + 11x^2 + 2x - 3$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 0, 2, 4, 2, 0, -1]), R([1, 1, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 0, 2, 4, 2, 0, -1], R![1, 1, 1, 1]);

sage: X = HyperellipticCurve(R([-3, 2, 11, 20, 11, 2, -3]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$800$$ $$=$$ $$2^{5} \cdot 5^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$8000$$ $$=$$ $$2^{6} \cdot 5^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$192$$ $$=$$ $$2^{6} \cdot 3$$ $$I_4$$ $$=$$ $$11604$$ $$=$$ $$2^{2} \cdot 3 \cdot 967$$ $$I_6$$ $$=$$ $$322392$$ $$=$$ $$2^{3} \cdot 3 \cdot 7 \cdot 19 \cdot 101$$ $$I_{10}$$ $$=$$ $$-1000$$ $$=$$ $$- 2^{3} \cdot 5^{3}$$ $$J_2$$ $$=$$ $$192$$ $$=$$ $$2^{6} \cdot 3$$ $$J_4$$ $$=$$ $$-6200$$ $$=$$ $$- 2^{3} \cdot 5^{2} \cdot 31$$ $$J_6$$ $$=$$ $$142400$$ $$=$$ $$2^{6} \cdot 5^{2} \cdot 89$$ $$J_8$$ $$=$$ $$-2774800$$ $$=$$ $$- 2^{4} \cdot 5^{2} \cdot 7 \cdot 991$$ $$J_{10}$$ $$=$$ $$-8000$$ $$=$$ $$- 2^{6} \cdot 5^{3}$$ $$g_1$$ $$=$$ $$-4076863488/125$$ $$g_2$$ $$=$$ $$27426816/5$$ $$g_3$$ $$=$$ $$-3280896/5$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

 This curve has no rational points. This curve has no rational points. This curve has no rational points.

magma: []; // minimal model

magma: []; // simplified model

Number of rational Weierstrass points: $$0$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

This curve is locally solvable except over $\Q_{2}$ and $\Q_{5}$.

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: $$\Z/{4}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + x^2z + xz^2 + z^3$$ $$0$$ $$4$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$5.590050$$ Tamagawa product: $$1$$ Torsion order: $$4$$ Leading coefficient: $$0.349378$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$5$$ $$6$$ $$1$$ $$1$$
$$5$$ $$2$$ $$3$$ $$1$$ $$( 1 - T )( 1 + T )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 20.a
Elliptic curve isogeny class 40.a

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.