The name and registry of the Cochrane have not been legible on screen. In the script of "The Drumhead", the Cochrane was only called a smaller ship and a Federation craft.  With the following appearance in "The Game", the unscripted vessel, was only referred to as a shuttle, and represented by stock footage from "The Drumhead". The Star Trek: The Next Generation Companion has revealed the ship was the Cochrane and according to Star Trek Encyclopedia it was also the Cochrane that transported Dax and Bashir to Deep Space 9 in "Emissary".
The Cochrane is mentioned in the backstory of Star Trek Online, which establishes that in 2396 the vessel's captain and two senior officers were found to be Undine spies and placed under arrest.