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The Elkies Curve has Rank 28 Subject only to GRH

In 2006, Elkies presented an elliptic curve with 28 independent rational points. We prove that subject to GRH, this curve has Mordell-Weil rank equal to 28 and analytic rank at most 28. We prove similar results for a previously unpublished curve of Elkies having rank 27.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, and 24. This complements the work of Jonathan Bober, who proved this claim subject to both the Birch and Swinnerton-Dyer rank conjecture and GRH. This gives some new evidence that the Birch and Swinnerton-Dyer rank conjecture holds for elliptic curves over Q of very high rank. Our results about Mordell-Weil ranks are proven by computing the 2-ranks of class groups of cubic fields associated to 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 to 22 and that the class group of a particular totally real cubic field has 2-rank equal to 20.