# Properties

 Label 5547.b.16641.1 Conductor 5547 Discriminant 16641 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: Magma / SageMath

## Minimal equation

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

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

$y^2 + y = x^6 - 3x^5 + x^4 + 3x^3 - x^2 - x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$5547$$ = $$3 \cdot 43^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$16641$$ = $$3^{2} \cdot 43^{2}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$2080$$ = $$2^{5} \cdot 5 \cdot 13$$ $$I_4$$ = $$100672$$ = $$2^{6} \cdot 11^{2} \cdot 13$$ $$I_6$$ = $$57396224$$ = $$2^{10} \cdot 23 \cdot 2437$$ $$I_{10}$$ = $$68161536$$ = $$2^{12} \cdot 3^{2} \cdot 43^{2}$$ $$J_2$$ = $$260$$ = $$2^{2} \cdot 5 \cdot 13$$ $$J_4$$ = $$1768$$ = $$2^{3} \cdot 13 \cdot 17$$ $$J_6$$ = $$16776$$ = $$2^{3} \cdot 3^{2} \cdot 233$$ $$J_8$$ = $$308984$$ = $$2^{3} \cdot 13 \cdot 2971$$ $$J_{10}$$ = $$16641$$ = $$3^{2} \cdot 43^{2}$$ $$g_1$$ = $$1188137600000/16641$$ $$g_2$$ = $$31074368000/16641$$ $$g_3$$ = $$126006400/1849$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2])

## Rational points

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);

This curve is locally solvable everywhere.

magma: [C![-5,-363,7],C![-5,20,7],C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-2,1],C![2,1,1],C![12,-363,7],C![12,20,7]];

Known rational points: (-5 : -363 : 7), (-5 : 20 : 7), (-1 : -2 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 1), (1 : 1 : 0), (2 : -2 : 1), (2 : 1 : 1), (12 : -363 : 7), (12 : 20 : 7)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank*: $$2$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.0062790848228 Real period: 24.460379506288139697494350989 Tamagawa numbers: 2 (p = 3), 1 (p = 43) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

### Sato-Tate group

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

### Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 129.a1
Elliptic curve 43.a1

### Endomorphisms

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 $$\Q$$.