  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 $d\leq 7$.
