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by QuesnayJr 583 days ago
You can get some higher degree examples (y squared = a degree 4 polynomial, for example), but degree 3 is special. An arbitrary polynomial of degree 4 and higher lack a rich structure (as far as we can tell). You can try to get around it by embedding the curve in a higher dimensional object, but it doesn't get you as far. (This is the idea behind hyperelliptic curve cryptography, for example.)