Parity of ranks for elliptic curves with a cyclic isogeny

Abstract: Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for , in particular confirming the parity conjecture for such curves. We prove the analogous results for under the additional assumption that E is not supersingular at primes above 2. [Copyright &y& Elsevier]