We don’t know much more than that in terms of where Fawkes came from.
Is it possible that Fantastic Beasts and Where To Find Them is going to reveal Fawkes’ origin story?
We know that Newt Scamander knew Dumbledore, and we know Dumbledore eventually comes into possession of a phoenix, which he is believed to have obtained at some point before 1938.
They think there’s a chance that the egg we see Jacob holding in the most recent trailer might be a phoenix egg.
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Harry Potter and the Chamber of Secrets Illustrated Edition Animated Book Trailer