TITLE: Is there a MAGMA function to calculate the absolutely irreducible components of an algebraic curve defined over the rationals?
QUESTION [8 upvotes]: Given a curve defined over the rationals, is it computationaly possible to find all its absolutely irreducible components?
Is there an implementation of this in the MAGMA program?

REPLY [3 votes]: An algorithm to find irreducible components is described here:
On computing absolutely irreducible components of algebraic varieties with parameters, by Ali Ayad (published in "Computing", 2010)
Abstract:
This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over Q, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is δO(r4)dr4dO(n3), being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters).
I don't know if this, or any other, algorithm is implemented in Magma, but I am guessing that a simple version is not so hard to implement.

REPLY [3 votes]: Magma (respectively, Sage!) can find the prime and primary components of a scheme $X$ (the irreducible components of a scheme $X$).
Look at the Magma documentation and also the Sage documentation on algebraic schemes. There are examples of usage in the documentation.