The department shall enter on the birth certificate of any child born out of wedlock the name of the father of the child upon receipt of (1) a certified copy of a court order showing that paternity has been established or a statement in writing by the father that he is the father of the child and (2) the written request of (a) the parent having legal custody of the child or (b) the guardian or agency having legal custody of the child. The surname of the child shall be determined in accordance with section 71-640.03.