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In 2006, Elkies presented an elliptic curve with 28 independent rationalpoints. We prove that subject to GRH, this curve has Mordell-Weil rank equal to28 and analytic rank at most 28. We prove similar results for a previouslyunpublished curve of Elkies having rank 27. We also prove that subject to GRH,certain specific elliptic curves have Mordell-Weil ranks 20, 21, 22, 23, and24. This complements the work of Jonathan Bober, who proved this claim subjectto both the Birch and Swinnerton-Dyer rank conjecture and GRH. This gives somenew evidence that the Birch and Swinnerton-Dyer rank conjecture holds forelliptic curves over Q of very high rank. Our results about Mordell-Weil ranksare proven by computing the 2-ranks of class groups of cubic fields associatedto these elliptic curves. As a consequence, we also succeed in proving that,subject to GRH, the class group of a particular cubic field has 2-rank equal to22 and that the class group of a particular totally real cubic field has 2-rankequal to 20.

The Elkies Curve has Rank 28 Subject only to GRH