2:26am

"...is neither 0 or 1728"; OK THANK you number theory

Alright. I've been ALL MORNING thinking about this so I had to do some search and I've realized 1728 is not that weird in elliptic curves - in fact it's pretty common. Incoming thread to feel that I haven't wasted my morning. (Please correct me if I'm saying something wrong!)

We are in a context of considering elliptic curves over a field. We are interested in counting rational points in elliptic curves (this is interesting not only in theory but for applications in cryptography). Starting from the beginning, there are several forms of writing ECs: