Epic or Elegy?

In my Anglo-Saxons history class that I have before this class we are discussing the Anglo-Saxon epic Beowulf. Something that we were discussing today really made me consider a similar question with regards to The Iliad. We were attempting to determine whether or not Beowulf was in fact an epic or if it was, in fact, and elegy. We didn't quite come up with a response but I began wondering if you could ask the same thing of The Iliad. Yes, The Iliad is the definition of an epic. The Oxford English Dictionary (OED) proves this when it describes an epic as "Pertaining to that species of poetical composition, represented typically by the Iliad and Odyssey, which celebrates in the form of a continuous narrative the achievements of one or more heroic personages of history or tradition." My question, however, is can you consider The Iliad as an elegy as well? Does it fit both of these categories? In my opinion, I'd have to say yes. Going back to the OED we can define an elegy as "A song or poem of lamentation, esp. for the dead; a memorial poem." This definition further enforces the idea that this poem is in fact an elegy of sorts as well as an epic. If Alice Oswald believes that she captured the essence of The Iliad in her Memorial than it seems that this proves that it is in fact an elegy. After all, and elegy is "a memorial poem". The next question this raises is how can these two different genres fit together? In our Anglo-Saxons class we read an article by J.R.R. Tolkien in which Tolkien seems to argue that Beowulf is a "heroic elegy". I believe we can apply this same type of spliced-genre to The Iliad as well. It is an epic which tells the story of heroes but it also elegiacally mourns the deaths of everyone at Troy.