  We generalize the lemmas of Thomas Kretschmer to arbitrary number fields, and
apply them with a 2-descent argument to obtain bounds for families of elliptic
curves over certain imaginary quadratic number fields with class number 1. One
such family occurs in the congruent number problem. We consider the congruent
number problem over these quadratic number fields, and subject to the
finiteness of Sha, we show that there are infinitely many numbers that are not
congruent over Q but become congruent over $Q(\zeta_3)$.
