Coated conductors are now being considered for use in beam steering and focusing magnets for high energy particle accelerators. However, coated conductors and cables have substantial magnetization, important for field quality. Magnetization decay, although suppressed, is also important for field quality retention. In this work we investigate the influence of applied current and selected field cycle on magnetization and magnetization decay for YBCO coated conductors at low temperatures. Field cycles mimicking what would be experienced for accelerators were applied to the samples, including (1) starting a a high field, (2) dropping to a field between 0-1 T, and (3) increasing again to a field of 1 T. The magnetization and magnetization decay were measured as a function of the lower field point, and it was seen that while M could be suppressed with a conductor-specific lower field, time dependence (decay) was only partially reduced. Additionally, current of 0-100 A were applied and the influence of the magnetization for fields applied perpendicular to the tape surface were mapped as a function of lower field hold point and injected current.