> P<x> := PolynomialRing(RationalField());
> C := HyperellipticCurve(x^5+x+5);
> Genus(C);
2
> J := Jacobian(C);
> RankBounds(J);
1 1

> ptsC := Points(C : Bound := 1000);
> I := ptsC[1]; I;
(1 : 0 : 0)

> TorsionSubgroup(J);
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse]

> ptsJ := Points(J : Bound := 1000); ptsJ;
{@ (1, 0, 0), (x^2 + x + 1, 2, 2), (x^2 + x + 1, -2, 2) @}
> O := ptsJ[1]; O;
(1, 0, 0)
> P := ptsJ[2]; P;
(x^2 + x + 1, 2, 2)

> Order(O);
1
> Order(P);
0

> heightconst := HeightConstant(J : Effort:=2, Factor);
> LogarithmicBound := Height(P) + heightconst;  // Bound on h(Q)
> AbsoluteBound := Ceiling(Exp(LogarithmicBound));
> PtsUpToAbsBound := Points(J : Bound:=AbsoluteBound );
> ReducedBasis([ pt : pt in PtsUpToAbsBound ]);
[ (x^2 + x + 1, 2, 2) ]
[1.91676497996095562630531698693]
> Height(P)
1.91676497996095562630531698693

> all_ptsC := Chabauty(P : ptC:=I); all_ptsC;
{ (1 : 0 : 0) }