First, I know there are differences in logical qubits and physical qubits. It takes more physical qubits for each logical qubit due to quantum error. 

Wikipedia states that it takes quantum **gates** of order $\mathcal{O}((\log N)^2(\log \log N)(\log \log \log N)$ using fast multiplication for Shor's Algorithm. That comes out to $1,510,745$ **gates** for $2^{1024}$. Further down the article, it says that it usually take $n^3$ gates for $n$ qubits. This would mean it would take ~$115$ qubits.

However, I've run Shor's Algorithm as implemented in Q# samples using Quantum Phase Estimation and it comes out to $1025$ qubits.