Recently I was considering cubic extensions $K/Q$ that have discriminant negative of a perfect square. Classifying these curves reduces to solving a Diophatine equation of the form $4a^3+27b^2=c^2$ which is not that difficult. However, another approach is to note that the unique quadratic subfield of the Galois closure of $K$ is $Q[i]$, hence we are looking for cubic cyclic extensions of $Q[i]$. We should be able to write these down by CFT, and in particular, they should all be subfields of $Q[i][E_{tors}]$ where $E:y^2=x^3+x$ (if I'm not mistaking). However, I'm having a hard time finding the correct torsion points that generate a field containing some of my K.

As a particular example, consider the field generated by the cubic polynomial $x^3-11x+194/3$. This field has discriminant $-324=-18^2$. However, as far as I can tell $Q[i][E[n]]$ does not contain this field for $n<30$. How can I find this value of $n$, or where am I going wrong?

Thanks, Soroosh