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We consider the problem of determining the density of primes $p$ for which $\\alpha \\in E(\\mathbb{F}_{p})$ has odd order. This density is determined by the image of the arboreal Galois representation $\\tau_{E,2^{k}} : {\\rm Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q}) \\to {\\rm AGL}_{2}(\\mathbb{Z}/2^{k}\\mathbb{Z})$. 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