# Properties

 Label 864.a.221184.1 Conductor 864 Discriminant -221184 Mordell-Weil group $$\Z/{12}\Z$$ Sato-Tate group $N(G_{1,3})$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\C \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{CM} \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + x^3y = x^5 - 4x^4 - 6x^3 + 33x^2 - 36x + 12$ (homogenize, simplify) $y^2 + x^3y = x^5z - 4x^4z^2 - 6x^3z^3 + 33x^2z^4 - 36xz^5 + 12z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 16x^4 - 24x^3 + 132x^2 - 144x + 48$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![12, -36, 33, -6, -4, 1], R![0, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([12, -36, 33, -6, -4, 1]), R([0, 0, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([48, -144, 132, -24, -16, 4, 1]))

## Invariants

 Conductor: $$N$$ = $$864$$ = $$2^{5} \cdot 3^{3}$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(864,2),R![1]>*])); Factorization($1); Discriminant: $$\Delta$$ = $$-221184$$ = $$- 2^{13} \cdot 3^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$2688$$ = $$2^{7} \cdot 3 \cdot 7$$ $$I_4$$ = $$8847360$$ = $$2^{16} \cdot 3^{3} \cdot 5$$ $$I_6$$ = $$-866009088$$ = $$- 2^{14} \cdot 3^{2} \cdot 7 \cdot 839$$ $$I_{10}$$ = $$-905969664$$ = $$- 2^{25} \cdot 3^{3}$$ $$J_2$$ = $$336$$ = $$2^{4} \cdot 3 \cdot 7$$ $$J_4$$ = $$-87456$$ = $$- 2^{5} \cdot 3 \cdot 911$$ $$J_6$$ = $$10192896$$ = $$2^{11} \cdot 3^{2} \cdot 7 \cdot 79$$ $$J_8$$ = $$-1055934720$$ = $$- 2^{8} \cdot 3^{2} \cdot 5 \cdot 71 \cdot 1291$$ $$J_{10}$$ = $$-221184$$ = $$- 2^{13} \cdot 3^{3}$$ $$g_1$$ = $$-19361664$$ $$g_2$$ = $$14998704$$ $$g_3$$ = $$-5202624$$

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

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

## 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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1)$$

magma: [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: $$\Z/{12}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$2 \cdot(1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z)^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - z^3$$ $$0$$ $$12$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$18.14296$$ Tamagawa product: $$3$$ Torsion order: $$12$$ Leading coefficient: $$0.377978$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$13$$ $$5$$ $$3$$ $$1$$
$$3$$ $$3$$ $$3$$ $$1$$ $$1 + T$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $N(G_{1,3})$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 36.a2
Elliptic curve 24.a2

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

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

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-3})$$ with defining polynomial $$x^{2} - x + 1$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an order of index $$8$$ in $$\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-3})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$