This online help node and the nodes below it describe the specialised categories of nonsingular plane curves of genus zero: The rational plane curves and conics. Rational curves and conics in Magma are nonsingular plane curves of degree 1 and 2, respectively. The central functionality for conics concerns the existence of points over the rationals. If a point is known to exist then a conic can be parametrised by a projective line or a rational curve. In addition, several algorithms are implemented to convert conics to standard Legendre, or diagonal, models and, for curves over Q, to a reduced Legendre model or a minimal model. A rational curve in Magma is a linearly embedded image of the projective line to which the full machinery of algebraic plane curves can be applied. The special category of conics is called CrvCon and that of rational curves is CrvRat.
The central algorithms of this chapter deal with the classification and reduction of genus zero curves to one of the standard models to which efficient algorithms can be applied. These special types serve to classify all curves up to birational isomorphism. Since the canonical divisor KC of a genus zero curve is of degree -2, a basis for the Riemann--Roch space of the effective divisor -KC has dimension 3 and gives an anti-canonical embedding in the projective plane. The homogeneous quadratic relations between the functions define a conic model for any genus zero curve. If the curve has a rational point then a similar construction with the divisor of this point gives a birational isomorphism with the projective line. For conics over the rationals, efficient algorithms of Simon [Sim05] allow one to first find a point, if one exists, and then to reduce to simpler models. The existence of a point is easily determined by local conditions, and this local data is carried by the data of the ramified or bad primes of reduction; if such a point exists then the existence can be certified by a certificate. Simon's algorithm in fact parametrises the curve (by the projective line) which gives a birational isomorphism with the curve.
Not every curve of genus zero can be "trivialised" by reduction to a rational curve in this way; the obstruction to having a rational point, and therefore to being parametrised by a projective line, is measured by the primes of bad reduction and also by the automorphism group, both of which are closely associated to an isomorphism class of quaternion algebras. The final algorithms of this chapter make use of this connection to compute the automorphism group of a curve and to find isomorphisms between conics.