Abstract: Let K be a field of characteristic zero and let ? be an algebraic element of degree n over K. Given a proper parametrization ? of a rational curve C with coefficients in K(?), we present a new algorithm to compute the hypercircle associated to the parametrization ?. As a consequence, we can decide if C is defined over K and, if not, we can compute the minimum field of definition of C containing K. The algorithm exploits the structure of the conjugate curves of C but avoids computing in the normal closure of K(?) over K.

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