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

The elliptic curve (Formula presented.) admits a natural 3-isogeny (Formula presented.). We compute the average size of the (Formula presented.) -Selmer group as (Formula presented.) varies over the integers. Unlike previous results of Bhargava and Shankar on (Formula presented.) -Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on (Formula presented.); this sensitivity can be precisely controlled by the Tamagawa numbers of (Formula presented.) and (Formula presented.). As a consequence, we prove that the average rank of the curves (Formula presented.), (Formula presented.), is less than 1.21 and over (Formula presented.) (respectively, (Formula presented.)) of the curves in this family have rank 0 (respectively, 3-Selmer rank 1).