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The Igusa local zeta function is a generating function which counts, for a fixed prime number p, the number of solutions of polynomial congruence f(x) ≡ 0 modulo p, p2, p3, and so on. Naturally, such a quantity bears deep relations to other important mathematical ideas from number theory, algebraic geometry and singularities theory. In this work we explore some computational aspects of the Igusa local zeta function associated to the nondegenerate plane cubics over ℚp for p≠ 2,3.