# Properties

 Label 1269.b.102789.2 Conductor 1269 Discriminant 102789 Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

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

$y^2 + xy = x^5 - x^4 + x^2 + x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1269$$ = $$3^{3} \cdot 47$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$102789$$ = $$3^{7} \cdot 47$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$960$$ = $$2^{6} \cdot 3 \cdot 5$$ $$I_4$$ = $$-20160$$ = $$-1 \cdot 2^{6} \cdot 3^{2} \cdot 5 \cdot 7$$ $$I_6$$ = $$-1774656$$ = $$-1 \cdot 2^{6} \cdot 3^{3} \cdot 13 \cdot 79$$ $$I_{10}$$ = $$421023744$$ = $$2^{12} \cdot 3^{7} \cdot 47$$ $$J_2$$ = $$120$$ = $$2^{3} \cdot 3 \cdot 5$$ $$J_4$$ = $$810$$ = $$2 \cdot 3^{4} \cdot 5$$ $$J_6$$ = $$81$$ = $$3^{4}$$ $$J_8$$ = $$-161595$$ = $$-1 \cdot 3^{5} \cdot 5 \cdot 7 \cdot 19$$ $$J_{10}$$ = $$102789$$ = $$3^{7} \cdot 47$$ $$g_1$$ = $$102400000/423$$ $$g_2$$ = $$640000/47$$ $$g_3$$ = $$1600/141$$
Alternative geometric invariants: G2

## Automorphism group

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

## 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![0,0,1],C![1,-2,1],C![1,0,0],C![1,1,1]];

All rational points: (0 : 0 : 1), (1 : -2 : 1), (1 : 0 : 0), (1 : 1 : 1)

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

Number of rational Weierstrass points: $$2$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 5 (p = 3), 1 (p = 47) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{10}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

### Decomposition

Simple over $$\overline{\Q}$$

### Endomorphisms

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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