Periodic continued fractions and elliptic curves over quadratic fields

Let f(x) be a square free quartic polynomial defined over a quadratic field K such that its leading coefficient is a square. If the continued fraction expansion of f(x) is periodic, then its period n lies in the set{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,22,26,30,34}. We write explicitly all such polynomials for which the period n occurs over K but not over Q and n∉{13,15,17}. Moreover we give necessary and sufficient conditions for the existence of such continued fraction expansions with period 13,15 or 17 over K.