One main distinction of K3 surfaces, among others, is that they form the only class of surfaces that might admit more than one elliptic fibration with section, which is not of product type [17, Lemma 12.18]. It is therefore a natural problem to classify such fibrations. This has been done in the past three decades, via different methods by several authors, see for instance [15,14,7,2,3,6] and [1]. Recently, the second and third authors have proposed a new method to classify elliptic fibrations on K3 surfaces which arise as double cover of rational elliptic surfaces. We refer the reader to [5] and [6] for more details.Let X be a K3 surface obtained as a double cover of an extremal rational elliptic surface defined over a number field k.