I should have looked at Henri Cohen's book on Computational Algebraic Number Theory first:
The algorithm in "Chapter 5 Algorithms for Quadratic Fields, §5.7 Computation of the Fundamental Unit" probably answers my question.
Still it might be interesting to know a priori which algorithms are used in the Pari functions. But of course that could vary from version to version ...
So how would I have to proceed to gain that knowledge?
Michael