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A racy and readable account of the present state of ignorance about elliptic curves defined over the rational field, with particular reference to the experimental work of Swinnerton-Dyer and the author [J. Reine Angew. Math. 212 (1963), 7--25; MR 26 #3669, and forthcoming papers], which have revealed what a rich structure of theory there must be, waiting to be discovered.
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