Rational Parametrizations of Algebraic Curves using a Canonical Divisor

For an algebraic curveCwith genus 0 the vector space (D) whereDis a divisor of degree 2 gives rise to a bijectivemorphismgfromCto a conicC2in the projective plane. We present an algorithm that uses an integral basis for computing (D) for a suitably chosenD. The advantage of an integral basis is that it contains all the necessary information about the singularities, so once the integral basis is known the (D) algorithm does not need work with the singularities anymore. If the degree ofCis odd, or more generally, if any odd degree rational divisor onCis known then we show how to construct a rational point onC2. In such cases a rational parametrization, which means defined without algebraic extensions, ofC2can be obtained. In the remaining cases a parametrization ofC2defined over a quadratic algebraic extension can be computed. A parametrizationofCis obtained by composing the parametrization ofC2with the inverse of the morphismg.