What is the participial phrase in the following sentence? Noted for her beauty, Helen of Troy was to become a well-known character in Greek mythology.

A participial phrase contains a participle, which is a verbal form with the endings - ed or - ing (or irregular, such as "written", as in "Written on monday (the piece was famous already on Tuesday)).

here the participle is "noted" and the whole phrase is this one:

Noted for her beauty