Sudden shutdown

I was doing some processor-intensive work on my recently purchased used Latitude e6440, and it suddenly simply shut down. Trying to restart it does nothing at all. I was using it on a docking station, have removed the battery, and connected the travel charger, and I get nothing at all. Doesn’t even pre-boot.

The computer is otherwise in great condition. What are my options here? Replace the motherboard? It that even possible?