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In number theory, the Elkies trinomial curves are certain hyperelliptic curves constructed by Noam Elkies which have the property that rational points on them correspond to trinomial polynomials giving an extension of Q with particular Galois groups.
Both of the above results are generalized by Leopoldt's "Spiegelungssatz", which relates the p-ranks of different isotypic components of the class group of a number field considered as a module over the Galois group of a Galois extension.