What evidence is there that Naruto isn't a descendant of the senju clan?

His last name is Uzumaki...duh.

Um.. Naruto is supposed to be named Naruto Namikaze after Minato's surname. But because of the nine-tails event and Naruto becoming the third kyubi host, the Third Hokage felt it best to name him Naruto Uzumaki instead. Also, Minato is not known for a clan. The manga hasn't shown any reveals that he belonged to any clan. But, because Kushina is in one and has mentioned her clan in manga 500, Naruto is a direct descendant of the Uzumaki clan.