Cubic post-critically finite polynomials defined over $\mathbb{Q}$

We describe and implement an algorithm to find all post-critically finite (PCF) cubic polynomials defined over $\mathbb{Q}$, up to conjugacy over ${\text{PGL}}_2(\bar{\mathbb{Q}})$. 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 $\mathbb{Q}$, we create a finite search space of cubic polynomials over $\mathbb{Q}$ that may be PCF. Using a computer search of these possibly PCF cubic polynomials, we find fifteen which are in fact PCF.