# Properties

 Label 1083.a.20577.1 Conductor 1083 Discriminant 20577 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![-3, 9, -13, 11, -5, 1], R![0, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 9, -13, 11, -5, 1]), R([0, 0, 0, 1]))

$y^2 + x^3y = x^5 - 5x^4 + 11x^3 - 13x^2 + 9x - 3$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1083$$ = $$3 \cdot 19^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$20577$$ = $$3 \cdot 19^{3}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$3616$$ = $$2^{5} \cdot 113$$ $$I_4$$ = $$218944$$ = $$2^{6} \cdot 11 \cdot 311$$ $$I_6$$ = $$293055488$$ = $$2^{10} \cdot 11 \cdot 26017$$ $$I_{10}$$ = $$84283392$$ = $$2^{12} \cdot 3 \cdot 19^{3}$$ $$J_2$$ = $$452$$ = $$2^{2} \cdot 113$$ $$J_4$$ = $$6232$$ = $$2^{3} \cdot 19 \cdot 41$$ $$J_6$$ = $$-8664$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 19^{2}$$ $$J_8$$ = $$-10688488$$ = $$-1 \cdot 2^{3} \cdot 19^{2} \cdot 3701$$ $$J_{10}$$ = $$20577$$ = $$3 \cdot 19^{3}$$ $$g_1$$ = $$18866536236032/20577$$ $$g_2$$ = $$30289293824/1083$$ $$g_3$$ = $$-1634432/19$$
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![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1]];

All rational points: (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (1 : 0 : 1)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank: $$1$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 3), 3 (p = 19) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

### 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 19.a2
Elliptic curve 57.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 $$\Q$$.