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Let K be a number field. For which primes p does there exist an elliptic curve E/K admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.

We present a Mordell-Weil sieve that can be used to compute points on certain bielliptic modular curves X 0 (N ) over fixed quadratic fields. We study , 61, 65, 79, 83, 89, 101, 131} and |d| < 100.