Veech groups for Origami-Teichmüller curves in specific strata of low genus

Teichmüller curves are algebraic curves in moduli space which are images
of Teichmüller disks, i.e. of isometrically and holomorphically embedded
Poincaré disks in Teichmüller space. A particularly handy class of
Teichmüller curves comes from so-called origamis (or square-tiled
surfaces). One starts with finitely many copies of the unit square in
the Euclidean plane and glues them along their boundary according to a
few combinatorial rules, such that one obtains a surface which is
endowed with a holomorphic differential or equivalently with a
translation structure. Teichmüller curves are up to birationality
determined by an associated discrete subgroup of SL(2,R) called the
Veech group. For origamis this is a subgroup of SL(2,Z) of finite index.
We study
for origamis in strata of low genus how far they are away from being
congruence groups.