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Non-Weierstrass curves

For the curve y = x + ax + bx + c (the general form of an elliptic curve with characteristic 3), the formulas are similar, with s = x + x x + x + ax + ax + b / y + y and x = s − a − x − x.

For a general cubic curve not in Weierstrass normal form, we can still define a group structure by designating one of its nine inflection points as the identity O. In the projective plane, each line will intersect a cubic at three points when accounting for multiplicity. For a point P, − P is defined as the unique third point on the line passing through O and P. Then, for any P and Q, P + Q is defined as − R where R is the unique third point on the line containing P and Q.

For an example of the group law over a non-Weierstrass curve, see Hessian curves.