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|Harry Potter and the Goblet of Fire
A Cockatrice is, in Muggle texts, a synonym for a Basilisk. Though never fully defined in the Harry Potter series, it is apparently a dangerous creature that reproduces by means of eggs.
There are only two mentions of a Cockatrice anywhere in the series. One is when Karkaroff is accusing Professor Moody of being over-cautious, perhaps to the point of paranoia. Karkaroff says that Moody had destroyed a gift given to him because it was ticking and he had thought it might contain a Cockatrice egg; in fact it was a carriage clock.
The other is when Hermione, in response to Ron's question about who the five judges for the Triwizard Tournament are, says that three of the judges must be the heads of the three schools. This was because she has found mention of the Triwizard Tournament of 1792, in which all three school heads were injured when a cockatrice the Champions were supposed to be catching went on a rampage.
Interestingly, cockatrices are not mentioned in "Newt Scamander"'s Fantastic Beasts and Where to Find Them.
From the limited description we have, we do not know whether Basilisks and Cockatrices are the same creature, as our Muggle literature would indicate. As a Basilisk hatches from a chicken's egg (in Muggle mythology, a rooster's egg) incubated by a toad, it is possible; but no mention is made of the basilisk egg making a ticking sound. One also would have to wonder if the egg could be taken away from the toad before it hatched: would the lack of incubation prove fatal to the basilisk?