# Properties

 Label 691.a.691.1 Conductor 691 Discriminant -691 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, 0, -1, -1, 0, 1], R![1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -1, -1, 0, 1]), R([1, 1]))

$y^2 + (x + 1)y = x^5 - x^3 - x^2$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$691$$ = $$691$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-691$$ = $$-1 \cdot 691$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$416$$ = $$2^{5} \cdot 13$$ $$I_4$$ = $$-13184$$ = $$-1 \cdot 2^{7} \cdot 103$$ $$I_6$$ = $$-1301312$$ = $$-1 \cdot 2^{6} \cdot 20333$$ $$I_{10}$$ = $$-2830336$$ = $$-1 \cdot 2^{12} \cdot 691$$ $$J_2$$ = $$52$$ = $$2^{2} \cdot 13$$ $$J_4$$ = $$250$$ = $$2 \cdot 5^{3}$$ $$J_6$$ = $$601$$ = $$601$$ $$J_8$$ = $$-7812$$ = $$-1 \cdot 2^{2} \cdot 3^{2} \cdot 7 \cdot 31$$ $$J_{10}$$ = $$-691$$ = $$-1 \cdot 691$$ $$g_1$$ = $$-380204032/691$$ $$g_2$$ = $$-35152000/691$$ $$g_3$$ = $$-1625104/691$$
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,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0]];

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

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: 1 (p = 691) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{8}\Z$$

### 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 $$\Q$$.