A soybean cultivar designated 15070051 is disclosed. The invention relates to the seeds of soybean cultivar 15070051, to the plants of soybean cultivar 15070051, to the plant parts of soybean cultivar 15070051, and to methods for producing progeny of soybean cultivar 15070051. 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. The invention also relates to soybean cultivars or breeding cultivars, and plant parts derived from soybean cultivar 15070051. The invention also relates to methods for producing other soybean cultivars, lines, or plant parts derived from soybean cultivar 15070051, 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 cultivar 15070051 with another soybean cultivar.