Just to add, in the key of E major you have E F# G# A B C# D# which in formula is 1 2 3 4 5 6 7. If you see in the scale the C# is the 6th note in this case. The 6th degree is always the relative minor of the major scale – whatever it is! The relative minor is known as Aeolian. Rooting from the 6th degree you get C# D# E F# G A# B which in formula is 1 2 b3 4 5 b6 b7. So though we have the same notes, our formula says otherwise. The E major scale would work in terms of soloing, but it would be more like a substitution. So you are right in terms of home key, but the correct key or mode would be C# Aeolian.