On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let $m\ge -1$ be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations $$X^3-m{X}^{2}Y-(m+3)X{Y}^2-Y^3=\lambda$$ where $\lambda >0$ is a divisor of $m^2+3m+9$ and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki’s result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors $\lambda$ of $m^2+3m+9$. As a consequence, we obtain another proof of Okazaki’s theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for $m=-1,0,1,2,3,5,12,54,66,1259,2389$.