Fast jets are thought to be a crucial ingredient of star formation because they might extract angular momentum from the disk and thus allow mass accretion onto the star. However, it is unclear whether jets are ubiquitous, and likewise, their contribution to mass and angular momentum extraction during protostar formation remains an open question. Our aim is to investigate the ejection process in the low-mass Class 0 protostar L1157. This source is associated with a spectacular bipolar outflow, and the recent detection of high-velocity SiO suggests the occurrence of a jet. Observations of CO 2-1 and SiO 5-4 at 0.8" resolution were obtained with the IRAM Plateau de Bure Interferometer as part of the CALYPSO large program. The jet and outflow structure were fit with a precession model. We derived the column density of CO and SiO, as well as the jet mass-loss rate and mechanical luminosity. High-velocity CO and SiO emission resolve for the first time the first 200 au of the outflow-driving molecular jet. The jet is strongly asymmetric, with the blue lobe 0.65 times slower than the red lobe. This suggests that the large-scale asymmetry of the outflow is directly linked to the jet velocity and that the asymmetry in the launching mechanism has been at work for the past 1800 yr. Velocity asymmetries are common in T Tauri stars, which suggests that the jet formation mechanism from Class 0 to Class II stages might be similar. Our model simultaneously fits the inner jet and the clumpy 0.2 pc scale outflow by assuming that the jet precesses counter-clockwise on a cone inclined by 73 degree to the line of sight with an opening angle of 8 degree on a period of 1640 yr. The estimated jet mass flux and mechanical luminosity are 7.7e-7 Msun/yr, and 0.9 Lsun, indicating that the jet could extract at least 25% of the gravitational energy released by the forming star.