common sense vs sagacity

common sense and sagacity both are nouns.

Word	Noun	Adjective	Verb	Adverb
common sense	Yes	No	No	No
sagacity	Yes	No	No	No

As nouns, sagacity is a hypernym of common sense; that is, sagacity is a word with a broader meaning than common sense:

common sense: sound practical judgment
sagacity: the mental ability to understand and discriminate between relations

Other hypernyms of common sense include discernment, judgement, judgment, sagaciousness.

common sense (noun)	sagacity (noun)
sound practical judgment	the trait of forming opinions by distinguishing and evaluating
	the mental ability to understand and discriminate between relations