Usually the URL is in the following form: https://<server and port if applicable>/rm/resources/<identifier for the artifact>

It will be identical to the URL that shows up when you go to the artifact and click the "Show Permalink" icon.

Hope this helps

bob

The proper way to interact using a REST API and getting artifacts using external apps and tools is using the Open Services for Lifecycle Collaboriation (OSLC) public API.

You can get started by looking at this Jazz article that links to the OSLC workshop, which is a great way to get going with the OSLC REST API.
The workshop document shows you how to navigate through the REST API and URL's; querying for artifacts, getting artifacts, updating artifacts, getting the project properties as which artifacts types are defined and their attributes etc.
The code samples are in Java, but are simple so should be easy to translate to C#.