'Integral basis nf.zk versus nfbasis in Pari GP

I have been using the database of lmfdb.org to know the integral basis of a number field. Now, I want to utilize Pari GP in multiplying algebraic integers. However, I have encountered a problem. Pari GP uses the integral basis "nf.zk" in its computations, which apparently is not always the same as the "nfbasis(f)", which is the integral basis that lmfdb.org provides.

For example, we have the following code from Pari GP.

? f = x^3 - x^2 + 2*x + 8
nf = nfinit(f)
nf.zk
%1 = [1, x, 1/2*x^2 - 1/2*x + 1]
? nfbasis(f)
%2 = [1, x, 1/2*x^2 - 1/2*x]

Now, my questions are:

  1. Why are nf.zk and nfbasis(f) different?
  2. Why does Pari GP use nf.zk instead of nfbasis(f)?
  3. Lastly, can I tell Pari GP to use nfbasis(f) instead of nf.zk?
paripari-gp


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