Irregular elliptic surfaces of degree 12 in the projective fourspace

So far only a few families of smooth irregular surfaces are known to exist in P^4 up to pullbacks by suitable finite morphisms from P^4 onto P^4 itself. In this paper we present two different constructions of irregular smooth minimal elliptic surfaces of degree 12 in P^4. The first is a monad construction while the other uses liaison. The family constructed via liaison includes the surfaces of the first family as a special case.