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

@article{Poonen2005TwistsOX,
  title={Twists of X(7) and primitive solutions to x^2+y^3=z^7},
  author={Bjorn Poonen and Edward F. Schaefer and Michael Stoll},
  journal={Duke Mathematical Journal},
  year={2005},
  volume={137},
  pages={103-158}
}
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… 

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