The Medusan vessel went unseen in the original airing of the episode, only being referenced in dialog. Indeed, the original ending sequence had the Enterprise approaching the Medusan homeworld. This was altered in the 2008remastered version of the episode, where a CGI model was created for the vessel.
Based on information obtained from Michael Okuda, no name or registry number was specifically assigned to the model. In terms of its design, Okuda added that "the basic idea was Dave Rossi's suggestion. He thought that because the Medusans were non-corporeal and might not have developed technology as we know it, they might use Federation hardware, so their ships could have a distinct Federation style."