Number of Abelian Cubic Number Fields of given Discriminant

Question

According to Cohn's paper on "the density of Abelian cubic fields", there are exactly $2^v$ Abelian cubic extensions of $\mathbb{Q}$ with discriminant $f^2$, where $v\ge0$ and $f$ is of the form $f=9p_1\cdots p_v$ or $f=p_0p_1\cdots p_v$ with $p_i$ being primes congruent to $1$ (mod $3$). I know the splitting field of an irreducible cubic polynomial is Abelian if and only if its discriminant is a square, but how can we deduce that it must be a product of primes as above and the exact number of extensions for every such $f$?