Calculation of Veech groups and Galois invariants of general origamis

Nontrivial examples of Teichm"uller curves have been studied systematically with notions of combinatorics invariant under affine homeomorphisms. An origami (square-tiled surface) induces a Teichm"uller curve for which the absolute Galois group acts on the embedded curve in the moduli space. In this paper, we study general origamis not admitting pure half-translation structure. Such a flat surface is given by a cut-and-paste construction from origami that is a translation surface. We present an algorithm for the simultaneous calculation of the Veech groups of origamis of given degree. We have calculated the equivalence classes, the $PSL(2,\mathbb{Z})$-orbits, and some Galois invariants for all the patterns of origamis of degree $d\le 7$.