The average size of the 3-isogeny Selmer groups of elliptic curves $y^2=x^3+k$

The elliptic curve $E_k\text{\hspace{0.17em}⁣}:y^2=x^3+k$ admits a natural 3-isogeny ${\varphi }_k\text{\hspace{0.17em}⁣}:E_k\to E_{-27k}$. We compute the average size of the ${\varphi }_k$-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 $E_k$ and $E_{-27k}$. As consequences, we prove that the average rank of the curves $E_k$, $k\in \mathbb{Z}$, is less than 1.21 and over $23\mathrm{\%}$ (resp. $41\mathrm{\%}$) of the curves in this family have rank 0 (resp. 3-Selmer rank 1).