# Properties

 Label 4400.b.352000.1 Conductor 4400 Discriminant -352000 Mordell-Weil group $$\Z \times \Z/{3}\Z \times \Z/{6}\Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 1, -3, -2, 0, 1]), R([]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 1, -3, -2, 0, 1], R![]);

sage: X = HyperellipticCurve(R([1, 3, 1, -3, -2, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$4400$$ $$=$$ $$2^{4} \cdot 5^{2} \cdot 11$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-352000$$ $$=$$ $$- 2^{8} \cdot 5^{3} \cdot 11$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-2464$$ $$=$$ $$- 2^{5} \cdot 7 \cdot 11$$ $$I_4$$ $$=$$ $$480256$$ $$=$$ $$2^{10} \cdot 7 \cdot 67$$ $$I_6$$ $$=$$ $$-525623296$$ $$=$$ $$- 2^{13} \cdot 11 \cdot 19 \cdot 307$$ $$I_{10}$$ $$=$$ $$-1441792000$$ $$=$$ $$- 2^{20} \cdot 5^{3} \cdot 11$$ $$J_2$$ $$=$$ $$-308$$ $$=$$ $$- 2^{2} \cdot 7 \cdot 11$$ $$J_4$$ $$=$$ $$-1050$$ $$=$$ $$- 2 \cdot 3 \cdot 5^{2} \cdot 7$$ $$J_6$$ $$=$$ $$416900$$ $$=$$ $$2^{2} \cdot 5^{2} \cdot 11 \cdot 379$$ $$J_8$$ $$=$$ $$-32376925$$ $$=$$ $$- 5^{2} \cdot 7 \cdot 17 \cdot 10883$$ $$J_{10}$$ $$=$$ $$-352000$$ $$=$$ $$- 2^{8} \cdot 5^{3} \cdot 11$$ $$g_1$$ $$=$$ $$984285148/125$$ $$g_2$$ $$=$$ $$-871563/10$$ $$g_3$$ $$=$$ $$-2247091/20$$

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

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

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

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: $$\Z \times \Z/{3}\Z \times \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - z^3$$ $$0.365678$$ $$\infty$$
$$D_0 - 2 \cdot(1 : 1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0$$ $$3$$
$$(-1 : -1 : 1) - (1 : 1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 2z^3$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.365678$$ Real period: $$17.62720$$ Tamagawa product: $$27$$ Torsion order: $$18$$ Leading coefficient: $$0.537157$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$8$$ $$9$$ $$1$$
$$5$$ $$2$$ $$3$$ $$3$$ $$( 1 - T )( 1 + T )$$
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 11 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 220.a4
Elliptic curve 20.a3

## 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 $$\Q$$.