The elliptic curve $E_k:y^2=x^3+k$ admits a natural 3‐isogeny ${\varphi }_k:E_k\to E_{-27k}$. 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 a consequence, we prove that the average rank of the curves $E_k$, $k\in \mathbb{Z}$, is less than 1.21 and over $23\%$ (respectively, $41\%$) of the curves in this family have rank 0 (respectively, 3‐Selmer rank 1).

中文翻译：

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椭圆曲线的3个同质Selmer组的平均大小y2 = x3 + k

椭圆曲线 ${Ë}_{ķ}：{ÿ}^2=X^3+ķ$ 承认自然的三同基因 ${\varphi }_{ķ}：{Ë}_{ķ}\to {Ë}_{-27ķ}$。我们计算出${\varphi }_{ķ}$‐塞尔默集团 $ķ$在整数上变化。不像Bhargava和Shankar在$ñ$‐Selmer组的椭圆曲线，我们表明该平均值对以下条件下的同余条件非常敏感 $ķ$; 可以通过Tamagawa数精确控制该灵敏度${Ë}_{ķ}$ 和 ${Ë}_{-27ķ}$。结果，我们证明了曲线的平均等级${Ë}_{ķ}$， $ķ\in \mathbb{ž}$，小于1.21以上 $23％$ （分别， $41％$）在该系列中的曲线的等级为0（分别为3-Selmer等级1）。