There are three big conflicts in the novel- When Carter and Percy fight an enchanted crocodile, when Sadie and Annabeth fight the god Serapis, and when all four of them team up to fight Setne. In the beginning of the novel, there is also some conflict between Percy and Carter as they don't yet trust each other, and Annabeth has an internal conflict after destroying the Dog of Future, wondering if she destroyed her own future.
Climax: Fighting Setne
This is the book's first big battle. Percy and Carter set aside their differences to face off against a giant crocodile which is then revealed to be the beginning of a much bigger scheme. This scene establishes character relationships and begins to hint at the bigger conflict coming ahead.
Falling Action: Defeating Setne
The book's second big battle pits Sadie and Annabeth against the evil Serapis. This scene reveals Setne as the organizer behind everything and also builds an alliance and friendly relationship between Sadie and Annabeth, including even more events that hint at what's coming ahead.
Resolution: Saying Goodbye
After deciding that the only way to defeat Setne is to combine their powers of Greek and Egyptian magic, the four heroes charge into the final battle brandishing their new skills. The action hits its peak and the characters have finally learned how to work together, relating back to the theme that you can't always do things alone.
After failing to defeat Setne the first time, Percy decides to host the vulture goddess, and Sadie turns Annabeth into a magician by telling her secret name. Percy flies away with Setne, and plunges into the ocean with him. Percy is strengthened by the ocean, and together with his friends they trap Setne inside a snowglobe.
After defeating Setne, the heroes decide what to do with his crown, and realizing the power that could erupt if someone ever found it, decide to seal it away, despite the fact that it could also make all of them much more powerful. This shows that the characters have completed their arcs, accepting the existence of other forms of magic but also realizing that foreign magic should not be messed with. The heroes part ways, agreeing to contact each other if they ever need to again.