The form y^2 = x^3 + bx + c prohibits intuition and probably follows from the general form for an algebraic expression: ax^n + bx^(n-1) + ... The whole idea of

The form y^2 = x^3 + bx + c prohibits intuition and probably follows from the general form for an algebraic expression: ax^n + bx^(n-1) + ... The whole idea of an algebraic expression is to relate x and y (for 2D). What's the simplest relation? y/x = r => we have a straight line with slope r. Let's add a bit more detail. (y-b)/(x-a) = r => we shift the origin to (a,b) and check how the line looks. These relations can also get "self-aware". This is similar to how a rate of change can be self-aware (ex: d/dx f(x) = x). The rate of water drain out from a tap at the bottom of the water tank would depend on the water height which itself affected by the drain out. Something like that. (y-b)/(x-a) = x => That is a "self-aware" relation, because we replaced r (a constant) with the variable x. So we have a parabola now. Let's change gears a bit. we will use higher degree for variables y^2/x^2 = r => For positive values of r, we have a pair of straight lines. Now, ((y-a)(y-b))/((x-c)(x-d)) = r => What happened? we get a conic. Let's make it self-aware ((y-a)(y-b))/((x-c)(x-d)) = x => There it is. A cubic equation Digging deeper, we see that the self-awareness makes the curve to bring the left-side parts to join at finite distance rather than at negative infinity. To understand this, consider the pair of straight lines (y^2/x^2 = r). You might see that it is a pair of straight lines crossing each other. But you can also look at two sharply bent lines touching each other at a point. a conic has a bit smoother turn.