Saved you a click:

A phoenix is an oscillator in Conway’s Life where every cell dies in every generation. The smallest example is Phoenix 1, which oscillates with period 2 and has a constant population of 12:

All known finite phoenices have period 2, and Stephen Silver proved in 2000 that there cannot exist a finite phoenix of period 3. Alex Greason more recently proved the non-existence of any phoenix (finite or infinite) with period 3 or 5.

From the title I thought this was going to be some nerdy math joke about phoenixes being reborn from the ashes through some periodic cycle.

Only... that's exactly what this is. Perhaps it isn't a "joke," but names in math are frequently chosen with whimsy.