The location of Starbase 4077
Starbase 4077 was a Federation outpost administered by Starfleet. This starbase was located in Sector 001, which was a region of space in the Alpha Quadrant. It was located near the Sol system, Orion Sector Tactical Command, Starbase 74, and Starbase 0834. (TNG: "Conspiracy", display graphic)
This starbase's location was depicted twice on the map.
According to the Star Trek Encyclopedia, after "The Neutral Zone" aired, it was decided that the number of starbases would be kept low (i.e. under 500). The only other starbases whose numbers were higher than 500 are Starbase 514, Starbase 515, Starbase 621, Starbase 718, and Starbase 0834.