how is the order calculated for Secp256k1?

I can't seem to find any info on how they determine the order of Secp256k1 or any other elliptic curve cryptography stuff. The only clue I've come across is that the order (n) is typically pretty close to the prime field (p). So we've got: p = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F n = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141 Looks like the order and the prime field are about 2^128 apart. So if someone tried to brute force n starting at p, it would take 2^64 attempts to land on n. Did they really brute force n or was there an easier way to figure it out?

https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves https://en.wikipedia.org/wiki/Counting_points_on_elliptic_curves#Schoof%E2%80%93Elkies%E2%80%93Atkin_algorithm