Weber's Theorem
If two curves of the same curve genus >1 are in rational correspondence, then that correspondence is birational.
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References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 135, 1959.Referenced on Wolfram|Alpha
Weber's TheoremCite this as:
Weisstein, Eric W. "Weber's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WebersTheorem.html