Given a sixth power free integer a, let Eₐ be the elliptic curve defined by y²=x³+a. We prove explicit results for the lower density of sixth power free integers a for which the $3$-isogeny induced Selmer group of Eₐ over Q(μ₃) has dimension ≤ 1. The results are proven by refining the strategy of Davenport--Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for $3$-isogeny induced Selmer groups.
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Pratiksha Shingavekar (2024) studied this question.
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