Why was the eagle the symbol of Rome?

The legendary Roman eagle, hovering over the victorious legions. A symbol of the power and majesty of the Eternal City. Why the eagle and not some other symbol? Not the Capitoline wolf, for instance? It nurtured Romulus and Remus, the founders of Rome, while the eagle doesn’t seem to have nurtured anyone in Roman official history. Nor even warned of enemies, like the famous Roman geese. Why such a part for this bird?

Legion standard with eagle. Source: wikipedia.org

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