A soybean cultivar designated 4405070 is disclosed. The invention relates to the seeds of soybean cultivar 4405070, to the plants of soybean 4405070, to plant parts of soybean cultivar 4405070 and to methods for producing a soybean plant produced by crossing soybean cultivar 4405070 with itself or with another soybean variety. The invention also relates to methods for producing a soybean plant containing in its genetic material one or more transgenes and to the transgenic soybean plants and plant parts produced by those methods. This invention also relates to soybean cultivars or breeding cultivars and plant parts derived from soybean variety 4405070, to methods for producing other soybean cultivars, lines or plant parts derived from soybean cultivar 4405070 and to the soybean plants, varieties, and their parts derived from use of those methods. The invention further relates to hybrid soybean seeds, plants and plant parts produced by crossing the cultivar 4405070 with another soybean cultivar.