What is the hairy mammoth scientific name?

The scientific name for the woolly mammoth is Mammuthus Primigenius. A partial taxonomy for mammoth is:

Kingdom: Animalia

Phylum: Chordata

Class: Mammalia

Order: Proboscidea (long snout)

Family: Elephantidae (elephants and mammoths)

Genus: Mammuthus

There are several species within the Genus:Mammuthus, one of which is the woolly mammoth.