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Birational Transformation

How do I apply a birational transformation x --> 1/x, then clear the denominators. Is there a way I could do that in SAGE?

x, y = QQ['x,y'].gens()
C = Curve(x^2+y^2)
# the code I'm looking for here

$x^2 + y^2 \mapsto 1/x^2 + y^2 \mapsto 1/x^2 + x^2y^2/x^2 \mapsto 1 + x^2y^2$