Discriminant of a basis

Question

I am trying to find the discriminant of the Q-basis $ ( 1,\sqrt[3]{3},\sqrt[3]{9}) $ . I want to do it by explicit matrix computations. I know that I have to find embeddings, but I think there is only one embedding, Because the minimal polynomial for each element has repeated roots, so the determinant is zero. I don't know what is my mistake. I was wondering if somebody help me.