1. Given a curve defined over the rationals, is it computationaly possible to find all its absolutely irreducible components?
2. Is there an implementation of this in the MAGMA program?



Is there a MAGMA function to calculate the absolutely irreducible components of an algebraic curve defined over the rationals?
There is a more-or-less 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 on-line 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. 


