Click on Evaluate below to set up all the functions:
Now you can below draw all the lattice for a given order:
How to use it?
In the 'basic' case (see the third parameter below), the polynomial $X^a(1-X)^b-1$ will have a factor of a given order if one of the below lattice contain the point $(a,b)$.
In the 'decomposed' case (with colors) it is the same thing but with the polynomial $X^a(1-X)^b-\alpha$.
Full details (for the basic case) here