On the number of elliptic curves with prescribed isogeny or torsion group over number fields of prime degree

Let p be a prime and K a number field of degree p. We count the number of elliptic curves, up to K-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup, or that it has an isogeny of prescribed degree. We also study the following question: for a given n such that |Y0(n)(Q)| > 0, does every elliptic curve over K with an n-isogeny have a quadratic twist with torsion Z/nZ as a subgroup? We prove that this is true only for the cases n = 2, 3, 4, 6 and for the pairs (n,K) = (11,Q(ζ11)) and (n,K) = (14,Q(ζ7)).