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.