MO question on K1 of an elliptic curve
MO comment: I believe that one does not know finite generation for these groups. As for explicit elements, K1(E)(2) is generated by K1(P) as P runs over the closed points of E. One has a homomorphism from Pic(E⊗F)⊗F∗ to K1(E)(2) for any finite extensions F of Q by taking the cup product followed by the norm. The union of the images is the whole group.
nLab page on K-theory of regular models