No, Annuminas was the Tower of the West on the shore of Lake Nenuial (north of the Shire, west of Fornost) and was the home of the Elendil's Chief Palantir of the North, until it was lost.
Virumor has the answer with Elostirion, the tallest and western-most tower of the White Towers located in the Tower Hills to the west of the Shire. He gets the pretty-bauble.
What color were Tom Bombadil's boots