Abstract: Let K be a characteristic zero field, let ? be an algebraic element over K and C a rational curve defined over K given by a parametrization ? with coefficients in K?(?). We propose an algorithm to solve the following problem, that is, a parametric version of Hilbert-Hurwitz: To compute a linear fraction u=at+b:ct+d such that ??(u) has coefficients over an algebraic extension of K of degree at most two and a conic K-birational to C. Moreover, if the degree of C is odd or ? is of odd degree over K, we can compute a parametrization of C with coefficients over K. The problem is solved without implicitization methods nor analyzing the singularities of C.

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