  We observe that there are elliptic curves over number fields all of whose
quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer
conjecture. We give a classification of such curves in terms of their local
behaviour, and characterise them in terms of the Galois action on the Tate
module. In particular, their existence shows that Goldfeld's conjecture does
not extend directly to elliptic curves over number fields.
