ABSTRACT
Suppose K is a field not of characteristic 2 or 3. Let a, b ∈ K satisfy 4a3+27b2 6= 0. Then an elliptic curve overK is the set of points (x, y) ∈ K×K satisfying
The condition 4a3 + 27b2 6= 0 is vital. The cubic polynomial
x3 + ax+ b
has discriminant −16(4a3 + 27b2), which is equal to zero if and only if the polynomial has equal roots. Requiring the discriminant to be non-zero means that all roots of the polynomial are distinct, and this guards against “singular points” of the curve. This particular form of an equation for an elliptic curve is called the Weierstrass form or sometimes the simplified Weierstrass form of the curve. There are many other equations that describe elliptic curves.
