I placed six points on the circumference of a circle such that the distance between any two of the points is an integer. What is the smallest such circle I could use?

What if each distance must be unique?

(In reply to

com by Pemmadu Raghu Ramaiah)

On the contrary, for any n>0, there is a circle that passes through more than n lattice points ("points of Integer coordinates" as you call them).

However, that doesn't have anything to do with this puzzle, as it is the distances, not the coordinates, that have to be integers.