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According to the Star Trek Encyclopedia (2nd ed., p. 513), the vessel was a Rigel-class (Rigel class at Memory Beta, the wiki for licensed Star Trek works) starship, with the registry NCC-62095. The Encyclopedia notes that the ship was named after famed Earth author, Leo Tolstoy.
Originally the Tolstoy was designated the Template:ShipClass USS Chekov and the filming model was labeled as such. A last minute post-production change resulted in a name change as to avoid reference to the original series character Pavel Chekov, deemed too "cute" for the moment in TNG: "The Best of Both Worlds, Part II". The Tolstoys designation as Rigel-class was a later attempt by Michael Okuda to remedy the discrepancy. At the time of filming of the episode the class designation was not in use as he remembered,"Rigel: I don't think we ever came up with a ship design specifically tied to that name." It is unclear if the renaming at the time meant that the filming model was intended to be the Tolstoy or that a reference was made to a ship not seen in the episode, though Okuda apparently proceeded eventually from the former assumption. He himself reinforced that notion when he gave a screen-used piece of wreckage he owned, in consignment to the Propworx's Star Trek auction of 4 June 2011, described as, "Irregular piece of the saucer section of the U.S.S. Chekov from the "starship graveyard" scene in the Star Trek: The Next Generation episode "The Best of Both Worlds, Part. II."" (Propworx Star Trek auction catalog, page 44) The piece, Lot 42, estimated at $100-$200 sold for $400.
According to the Star Trek: New Frontier short story, "Performance Appraisal" in No Limits, Captain Christine Parsons (Christine Parsons at Memory Beta, the wiki for licensed Star Trek works) took command of the Tolstoy in 2366.
The Price of Freedom, sourcebook, describes the Tolstoy as a heavy scout with a crew of 90. It had 6 decks and was 215 meters in length. The ship had a maximum speed of warp factor 9.2. The ship was armed with type VII phaser emitters, and two torpedo launchers with a normal loadout of 75 photon torpedoes.