Rational Curves and Conics

There are two parametrisation algorithms for rational curves. The first is assumes you have a rational point in hand. The second makes the anticanonical embedding whose image is a conic curve which may or may not have points. When defined over the rational numbers, we now have a fast point finding and parametrisation algorithm for conics which have a rational point. For example, it takes 12 seconds to find a point (with reduced coordinates) on each of 100 curves of the form ax² + by² + cz² = 0 where a, b, care prime numbers of size around 10¹00. Finding a point on a single curve of the same form with coefficients of size around 1000 digits takes 24 seconds.

• A data type for rational curves
• A parametrisation algorithm for rational curves given a rational point
• A canonical parametrisation algorithm reducing rational curves to conics irrespective of the existence of a point
• A data type for plane conics
• A new algorithm for finding a point with small coefficients over the rationals if one exists
• Type translation from curves of genus 0 to rational curves