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Twists of X(7) and primitive solutions to x^2+y^3=z^7

Bjorn Poonen·Edward F. Schaefer·Michael Stoll
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Abstract

We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant twists, we exploit the isomorphism between X and the modular curve X(7), and use modularity of elliptic curves and level lowering. This leaves 10 genus-3 curves, whose rational points are found by a combination of methods.

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Bjorn Poonen, Edward F. Schaefer, Michael Stoll. 2005-08-10. Twists of X(7) and primitive solutions to x^2+y^3=z^7. https://doi.org/10.1215/s0012-7094-07-13714-1

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