An elliptic curve is the set of solutions to an equation of the form y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6. By changing variables, y->2y + a_1 x + a_3, assuming the field characteristic is not 2, the equation becomes y^2 = 4x^3 + b_2 x^2 + 2b_4 x + b_6 where b_2 | = | a_1^2 + 4a_2 b_4 | = | 2a_4 + a_1 a_3 b_6 | = | a_3^2 + 4a_6.

algebraic geometry | elliptic curve | Frey curve | isogeny | Weierstrass form