This is a study of square-tiled translation surfaces in the stratum
*H*(2) and their SL(2 , **R**)-orbits or Teichmüller discs, which
are arithmetic. We prove that for prime *n* > 3 translation surfaces
tiled by *n* squares fall into two Teichmüller discs, only
one of them with elliptic points, and that the genus of these discs
has a cubic growth rate in *n*.

**Keywords:** Teichmüller discs, square-tiled surfaces, Weierstrass
points.

**MSC:** 32G15 (37C35 30F30 14H55 30F35)