Interestingly enough, ChatGPT started his answer like this only once:

"This is exactly the question I would ask next. My impression is: Most standard invariants are...."

And this was a response to this prompt:

"Is there a chance of an indirect argument of X ~ A^3 coming from computing some invariant of X that forces it to be A^3? (I am not all that expert in algebraic geometry but I'm thinking like degree or Betti numbers or something.)"