Snooping around my router (which is a linksys wrt1900ac) it has Domain Name: cable.virginmedia.net under router info. I used to work for Virgin Media Digital Bank, though not sure why that would overwrite my router info?!
Not important now, however, it would be good to know about your previous comment around the modem mode and the end point/domain being a virgin suffix.
Just as with a website address is both a name you type in to a browser and an IP address, the same is true for the IP addresses assigned to end user kit.
When an IP address is assigned, it's at the last point that is designed to accept that assignment. A modem is designed to be invisible to an IP address assignment as it's not an end device in itself, whereas a router or computer is an end device.
As you're running in modem mode, it's your own router that's given the public IP address, and that public IP address also has a hostname that can identify where and on what bit of kit you're connected to the Virgin Media network.