Rational parametrizations of algebraic curves using a canonical divisor

Abstract

For an algebraic curve C with genus 0 the vector space L(D) where D is a divisor of degree 2 gives rise to a bijective morphism g from C to a conic C2 in the projective plane. We present an algorithm that uses an integral basis for computing L(D) for a suitably chosen D. 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 L(D) algorithm does not need work with the singularities anymore. If the degree of C is odd, or more generally, if any odd degree rational divisor on C is known then we show how to construct a rational point on C2. In such cases a rational parametrization, which means defined without algebraic extensions, of C2 can be obtained. In the remaining cases a parametrization of C2 defined over a quadratic algebraic extension can be computed. A parametrization of C is obtained by composing the parametrization of C2 with the inverse of the morphism g. © 1997 Academic Press Limited.