A computational approach to Hilbert modular group fixed points

Some useful information is known about the fundamental domain for certain Hilbert modular groups. The six nonequivalent points with nontrivial isotropy in the fundamental domains under the action of the modular group for $\mathbf{Q} ( \sqrt 5 )$, $\mathbf{Q}( \sqrt 2 )$, and $\mathbf{Q} ( \sqrt 3 )$ have been determined previously by Gundlach. In finding these points, use was made of the exact size of the isotropy groups. Here we show that the fixed points and the isotropy groups can be found without such knowledge by use of a computer scan. We consider the cases $\mathbf{Q} ( \sqrt 5 )$ and $\mathbf{Q} ( \sqrt 2 )$. A computer algebra system and a C compiler were essential in perfoming the computations.