Riddle for Zoom ID

Some board games require rolling two standard dice and adding the number of eyes obtained. This produces a number between 2 and 12 with a non-uniform probability distribution (e.g. P(2)=1/36, while P(7)=1/6). There is exactly one other way of labelling two blank dice with integers to produce the same probability distribution. Find this labelling.

Hint: Use polynomials and roots of unity.

Zoom ID = Multiply largest label on each die and add the product to 91757865310. (So for the standard labelling you would get 6*6+91757865310.)