Ray class field of $K=Q (\zeta_3)$ of conductor 6.

Question

I am studying class fields generated as Kummer extensions and I have studied decomposition laws in Kummer extensions. My particular example is $L=K(\sqrt[3]2 )$, where $K$ contains primitive cube roots of unity. $L$ is an abelian extension of $K$. I am not able to see how $L$ is the ray class field of $K$ of conductor $6$ (though I have factorized primes in the intermediate extension fields of $Q$ ).i.e . in $K=Q(\zeta_3)$ and in $M=Q(\sqrt[3]2)$. Thanks in advance.