Cubic postcritically finite polynomials defined over q

We find all post-critically finite (PCF) cubic polynomials defined over Q, up to conjugacy over PGL2(Q̄). We describe normal forms that classify equivalence classes of cubic polynomials while respecting the field of definition. Applying known bounds on the coefficients of post-critically bounded polynomials to these normal forms simultaneously at all places of Q, we create a finite search space of cubic polynomials over Q that may be PCF. Using a computer search of these possibly PCF cubic polynomials, we find fifteen which are in fact PCF.