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?



There is a moreorless standard algorithm for finding irreducible components. It is not necessarily quick and is a topic of current research to improve it or find alternatives. It goes as follows. First, find a smooth point on the curve. E.g. pick one coordinate at random and solve for the others. (Note you most likely will not get a rational point, just an algebraic point.) Next, compute many terms of the power series expansions, in terms of one coordinate, for the other coordinates around the point you found (e.g. using Hensel's lemma). Finally, using linear algebra, find the minimal degree equations satisfied by these power series. You have found the irreducible component passing through that point you found in the first step. Repeat. I am not sure if this is implemented in Magma. They have an extensive online list of functions. Have you looked there? Alternatively, contact them. I assume that the license you paid for entitles you to costumer support. I know something like that is implemented in Singular, which is used by Sage and that Sage was trying to come up with its own version. 


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 ndimensional 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. 


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. 

