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One thing I've always wondered about elliptic curves is why everything is so centered on degree 3 two variable polynomials.

Aren't there rich structures to be explored for curves of degree >3 ?

Or is 3 really special ?

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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.)

I think part of the reason why 3 is special is because you get a lot of bang for your buck. Order 3 is a low order polynomial that is relatively easy to analyse, but already gives tremendous mathematical properties.

For example, the points of elliptic curves form groups. The operation of combining the points is described in the article (draw a straight line through two points and mirror in x-axis).

That means that all the theorems that are proven for Groups, are also true for elliptic curves.

But I think there are many more exciting properties

Amateur here (just studying abstract algebra for hobby). I’m also very curious for more reasons.

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