Phoenix

Alternate Time line
The Phoenix was the fifth Daedalus-Class Warship built in an alternate timeline and was given to Colonel Carter to support the Atlantis Expedition. According to Dr. McKay, the ship was rushed into service and many of the Asgard systems weren't fully operational. Many of the Expedition members worked to get the ship fully functional. The ship featured an Asgard terminal on the bridge where the star map should have been, and a room similar to the Asgard core room on board the Odyssey. The Phoenix was introduced in "The Last Man"

Michael's Fleet
Once the ship was ready, Colonel Carter flew the ship and tried to destroy Michael's fleet. Using guerrilla warfare, Carter succeeds in destroying many of Michael's Hive ships. Unfortunately, Michael put out false intel and ambushed the Phoenix. When it was badly damaged, the crew was beamed down to the surface of a planet by Carter while she rams the badly damaged Phoenix into one of the Wraith ships. The resulting explosion destroys the Phoenix and 3 of Michael's hives.

Despite this being an alternate timeline, it is unknown if the ship is under construction in the current timeline.