It does not have a known name apart from 'trident'.
the function of Poseidons temple was for Poseidons honor.
He hasn't got one in mythology, merely titles and epithets.
He had no last known location because he was not real. Even if he were, he was a god, and thus, omnipresent.
Rhea.
cronus
Cronus and Rhea
Neptune
a trident
Triton
Rhea.
In myths, he does not have one.