How many rings? Quite a lot.
I know that's not a good answer. I might have to see if I can estimate what's in mine. I estimated that my entire aluminum set (hauberk + coif) is on the order of about 25,000 links, if I remember correctly.
Thought experiment: If the coif is about 25% the weight of the hauberk, the ratio would be 4:1, so 1/5 of the total... which would be about 5,000 links. That number sounds a little high. But I'm just guessing on the ratio at the moment, and in my case the coif has a particularly large mantle. I may have to take a close look at it and see if I can make a better estimate.