Thermopylae is propably the thing the SPartans are most famous for. The modern image of Sparta is heavily influenced by this battle, how did is shape the Spartans' own identity?
Everything you say prior to the comma is far truer than you know, and indeed, it's not just the modern image. u/Iphikrates has a most excellent post covering this matter, and I highly commend his flair profile if you want to know more than you ever thought you'd want to know about Ancient Greek warfare.