Elliptic Curve

Defining Equation

An elliptic curve E over a field F is defined by a Weierstrass equation

$$E/F : y^2+ a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$$

((1))

with $a_1,a_2,a_3,a_4,a_6 \in F$ and $\Delta \neq 0$, where Δ is the discriminant of E and is defined as follows:

$$\left.\begin{aligned}\Delta &= -d_2^2d_8 - 8d_4^3 - 27d_6^2 + 9d_2d_4d_6 \\ d_2 &= a_1^2 + 4a_2 \\ d_4 &= 2a_4 + a_1a_3 \\ d_6 &= a_3^2 + 4a_6 \\ d_8 &= a_1^2a_6 + 4a_2a_6 - a_1a_3a_4 + a_2a_3^2 - a_4^2.\end{aligned}\right\}$$

((2))

If L is any extension field of F, then the set of L-rational points on E is

where ∞ is the point at infinity.

Two elliptic curves E ₁ and E ₂ defined over F and given by Weierstrass equations (1...