I've made some slight crew changes. Just so its not the one we know.
Name of Ship:BC-303 Prometheus
Home Universe/Name of Franchise: Prometheus Url
Ship Class:BC-303 prototype. As above.
Role: 'Battlecruiser'
Physical Description/Image:

Physical Size (roughly):
Armament/Complement (roughly): 195Mx80Mx65M
History of the Ship:Prometheus is the first spaceship constructed by Earth capable of defending Earth and exploring worlds throughout the Galaxy outwith the Stargate Network.

It successfully protected SG-1 above Antartica against Anubis during his attack on Earth. Following the battle Colonel Pentergast took command of the Prometheus and due to the unreliable hyperdrive technology it became locked in orbit to perform patrols for enemy vessels. After SG-1 saved the Asgard it was refit with a new Asgardian Hyperdrive, Beaming Technology, and shields. Since then (in this reality) it has gone on several successful missions to Atlantis in the Pegasus Galaxy, as well as helping the newer BC-304 class protect Earth and her allies at home in the Milky Way.
Notable Crew:
Brigadier General Jack O'Neill
Major Erin Gant
Liuetenant Kevin Marks
Dr.Lindsey Novak

Also aboard are the members of SG-1.

Colonel Samantha Carter
Teal'c
Doctor Daniel Jackson
Major Cameron Mitchell

Optional, but helpful:
Unique Technology:
Asgard Transporter
Naqudah Generator
Transporter Rings
Crew Size: 115
Faction/Operator/Owner:

Name of Ship: Mother of Invention
Home Universe/Name of Franchise: Red Vs Blue / Halo
Ship Class:Paris Class Heavy Frigate
Role: Mobile Command center
Physical Size (roughly): 535M X 119M X 112M
Armament/Complement (roughly): 1x MAC Canon, 50x Archer Missile Pods, 12 Nuclear missile silos, 12 PDS guns.
History of the Ship: The Mother of Invention saw very little action during the Human-Covenant War. In the days after during the Remnant Wars it was assigned to Doctor Leonard Church to use as a base of operations for his 'Project Freelancer'. In the waning days of the war everyone was looking for the magic bullet. Doctor Church believed it was the use of humanity's greatest weapon, A.I.

As Doctor Halsey fell from grace and various other programs filled the void Project Freelancers concept was to take a group of extraordinary soldiers, give them armor enhancements with unique technology even better than the average Spartan, and then pair them with a unique A.I. Doctor Church however could only get one A.I, one made off himself. As a by-product, a fragment personality off his deceased wife also came along, which gave Doctor Church the idea to split the Alpha into various fragments. Before he could do so however he was given the opportunity to conduct his experiments and test his theories with impunity when Daniel Jackson appeared, recruiting the Mother of Invention to stop a threat known as the Replicators in another universe.

Notable Crew:
Doctor Leonard Church: Director of the Project.
Alpha: A.I of the Director. His existence is top secret.
Chairman: He is responsible for the Agents of Project Freelancer, their battle readiness his his priority.
Beta/Agent Texas: The first fragment, personality based on the Directors dead wife. Only the Director and Chairmen know its an A.I fragment piloting a robotic body.

Agents of Project Freelancer: I'm not going to list them all just now, but I'll keep a note off them as we go.

Optional, but helpful:
Unique Technology: Project Freelancer has an elite group of Spartan Warriors with unique armor enhancements. It also possesses one fully functional A.I (Alpha) and one fragment (Beta) which operates a robotic exoskeleton as 'Agent Texas'.
Crew Size: 320
Faction/Operator/Owner: Project Freelancer