contacting apple support when the device is dead and you do not have the serial number

Hello, iPhone died suddenly. Upon trying to get help from Apple support, I discovered that I need the serial number...that is only available on the dead phone. Aside from "borrowing another device, prying the sims card out of my phone, putting it in another phone to get this info" (really, that's the best I've seen, how can I start an apple support chat session?