We present an end-to-end trainable wavelet video coder based on motion compensated temporal filtering (MCTF). Thereby, we introduce a different coding scheme for learned video compression, which is currently dominated by residual and conditional coding approaches. By performing discrete wavelet transforms in temporal, horizontal, and vertical dimension, we obtain an explainable framework with spatial and temporal scalability. We focus on investigating a novel trainable MCTF module that is implemented using the lifting scheme. We show how multiple temporal decomposition levels in MCTF can be considered during training and how larger temporal displacements due to the MCTF coding order can be handled. Further, we present a content adaptive extension to MCTF which adapts to different motion strengths during inference. In our experiments, we compare our MCTF-based approach to learning-based conditional coders and traditional hybrid video coding. Especially at high rates, our approach has promising rate-distortion performance. Our method achieves average Bj{\o}ntegaard Delta savings of up to 21% over HEVC on the UVG data set and thereby outperforms state-of-the-art learned video coders.