(1701) The registry number (1701) of the ship was chosen to be easily visible at a glance by television viewers.

Fictional Star Trek History
In the fictional universe of Star Trek, the USS Enterprise (NCC-1701) was the ship in the Original NBC TV series Star Trek,which chronicled the vessels most famous assignment, a five year mission of exploration of diplomacy under Captain James T. Kirk’s command. It is the first Federation starship to bear the name Enterprise. Following the destruction of the NCC-1701, Kirk & his officers
were instrumental in saving Earth from a mysterious probe (as chronicled in Star Trek IV: The Voyage Home). As a reward, Kirk (who was demoted to captain as punishment for disobeying
orders & taking the NCC-1701 to the Genesis Planet) was given command of a new U.S.S. Enterprise, a virtually identical (from the outside)) vessel with the registry number NCC-1701A. This began a long tradition of continuing the NCC-1701 registry number through successive generations of Enterprise which in some timelines continues until at least the 26th Century.