The average size of the 3-isogeny Selmer groups of elliptic curves y2 = x3 + k

The elliptic curve Ek:y2=x3+k admits a natural 3-isogeny phi k:Ek -> E-27k. We compute the average size of the phi k-Selmer group as k varies over the integers. Unlike previous results of Bhargava and Shankar on n-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on k; this sensitivity can be precisely controlled by the Tamagawa numbers of Ek and E-27k. As a consequence, we prove that the average rank of the curves Ek, k is an element of Z, is less than 1.21 and over 23% (respectively, 41%) of the curves in this family have rank 0 (respectively, 3-Selmer rank 1).