J
The one supplied by your internet provider?
If you wish to use your own, best to check and see what equipment specs are compatible with your ISP and if they will allow you to use your own modem to connect to their network.
I run my own router as well. The OP is asking about a modem though. I have to assume they are referring to the equipment that bridges the cable coming out of the wall to the router that segments the local area network.
Two completely different pieces of equipment although modems supplied by the ISP these days usually do contain a cheapy builtin router.
To use a privately owned modem, the ISP will need to program their server to recognize the device and talk to it. Thus the need to ensure what one wishes to use is compatible and the ISP is prepared to allow it to be used.
Your router doesn't connect to the ISP, it connects to your cable/Fiber modem which is the equipment that connects directly to the ISP's network. If the ISP doesn't recognize the modem, then no internet service even if you have correctly configured a router. In this case all you'll have is a functioning LAN (if you have more than one computer) with no external access to anything not connected to the router either by ethernet or wifi.
No pettiness. My system is completely different than the one you described.lj we know you are special. You pettiness is telling. Sorry for whatever I said to become the focus of your ire.