The field and curve have to admit a primitive root of unity in order to make the efficient endomorphism work, so that's going to be part of the selection criteria for any of the K curves.
primitive root ?
Now I understand. It is slightly more complicated than that.
Private keys are not multiple of 18051648. Private keys are power of z mod q, where z is a primitive root mod q.
source
https://github.com/cysecud/ecc_weak_keys/issues/2#issuecomment-2508282587primitive root generate vulnerable subgroups what provide way to breack 256 bit key in seconds.On github more info about it.