2012 AMC 12A Problem 20

Consider the polynomial

P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)

The coefficient of x^{2012} is equal to 2^a. What is a?

\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 24

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.