This is a translation of Klein’s Ueber die Transformation siebenter Ordnung der elliptischen Funktionen, first published in Mathematische Annalen 14 (1879), 428–471, and dated early November 1878. It follows the text printed in his Gesammelte Mathematische Abhandlugen, except where typos not present in the original had crept into formulas. I redrew all the figures (they had already been redrawn for the Abhandlugen: see caption on page 320), except for Figure 9 and the bottom figures on pages 315 and 316. I have not attempted to modernize the terminology, except on a few occasions when the use of current language allowed me to replace a longwinded phrase by something crisper and clearer. Nor have I tried to approximate the English mathematical style of the time. The goal has been to produce a readable translation, as close to the original ideas as possible. Bibliographic citations have been converted to the house format, the editors of the Abhandlugen having taken similar liberties. Brackets, if not delimiting bibliographic tags, indicate interpolated text, written either for the Abhandlugen (unsigned, or K. = Klein, B.-H. = Bessel-Hagen) or for this edition (L. = Levy). I’m grateful to Jeremy J. Gray for many excellent suggestions. In the study of the fifth-order transformation of elliptic functions we encounter, along with the modular equation of sixth degree and its well-known resolvent of fifth degree, the Galois resolvent of degree 60, called the icosahedral equation, which governs both. Starting from the icosahedral equation one sees with great ease the rule of formation and the properties of those lower-degree equations. In this work I would like to further the theory of the transformation of the seventh order up to the same point. I have already shown in [Klein 1879a] how one can construct the modular equation of degree eight in its simplest form in terms of this theory. The corresponding resolvent of seventh degree was considered in [Klein 1879b]. The question now is to construct the corresponding Galois resolvent of degree 168 in a suitable way, and to derive from it those lower-degree equations. As is well-known, the root η of this Galois resolvent, regarded as a function of the period ratio ω, has the characteristic property of remaining invariant unde
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