Weierstrass Equations

Generalized Weierstrass equation

An elliptic curve $E$ over a field $\mathbb{K}$ is defined by the equation of the smooth Weierstrass curve

$$E: y^{2}+a_{1} x y+a_{3} y=x^{3}+a_{2} x^{2}+a_{4} x+a_{6},\quad\quad(1),$$

where $a_{1}, a_{2}, a_{3}, a_{4}, a_{6} \in \mathbb{K}$ and $\Delta \neq 0$, where $\Delta$ is the discriminant of $E$ and is given by

$$\left.\begin{array}{l} \Delta=-b_{2}^{2} b_{8}-8 b_{4}^{3}-27 b_{6}^{2}+9 b_{2} b_{4} b_{6} \\ b_{2}=a_{1}^{2}+4 a_{2} \\ b_{4}=2 a_{4}+a_{1} a_{3} \\ b_{6}=a_{3}^{2}+4 a_{6} \\ b_{8}=a_{1}^{2} a_{6}+4 a_{2} a_{6}-a_{1} a_{3} a_{4}+a_{2} a_{3}^{2}-a_{4}^{2} \end{array}\right\}.$$

The set of solutions is the points $(x, y)$ on $E$ that satisfy equation (1) together with an extra defined point $\mathcal{O}$ at infinity:

$$E(\mathbb{K})=\left\{(x, y) \in \mathbb{K} \times \mathbb{K}: y^{2}+a_{1} x y+a_{3} y-x^{3}-a_{2} x^{2}-a_{4} x-a_{6}=0\right\} \cup\{\mathcal{O}\}.$$

(Darrel Hankerson et al. (2006)Darrel Hankerson, Alfred J. Menezes, and Scott Vanstone. Guide to Elliptic Curve Cryptography. Springer Professional Computing. New York, 2006. Springer., Lawrence C. Washington. (2008)Lawrence C. Washington. Elliptic Curves: Number Theory and Cryptography, Second Edition. Discrete Mathematics and Its Applications. New York, 2008. CRC Press. and Annette Werner. (2013)Annette Werner. Elliptische Kurven in der Kryptographie. Springer-Lehrbuch. Berlin Heidelberg, 2013. Springer.).

Note: The condition $\Delta \neq 0$ ensures that the elliptic curve is smooth, i.e., has no singular points at which the curve has a cusp or more than one tangent line, as shown in the example below.

Example:

The two curves are shown in Figures 1 and 2 for $\mathbb{K}=\mathbb{R}$ depict two examples of singular Weierstrass curves, whereas smooth curves can be seen in figures 3 and 4.

Figures 1 and 2

• Figure 1: Graph of the curve $y^{2} = x^3$.
• Figure 2: Graph of the curve $y^{2}=x^{3}+x^{2}$ with (dashed) tangent lines at the origin.