Groups Attached to Elliptic Curves

Definition 10.1.1 (Elliptic Curve)   An elliptic curve over a field $K$ is a genus one curve $E$ defined over $K$ equipped with a distinguished point $\O\in E(K)$.

We will not define genus in this book, except to note that a nonsingular curve over $K$ has genus one if and only if over  $\overline{K}$ it can be realized as a nonsingular plane cubic curve. Moreover, one can show (using the Riemann-Roch formula) that over any field a genus one curve with a rational point can always be defined by a projective cubic equation of the form

$$Y^2 Z + a_1 XYZ + a_3 YZ^2 = X^3 + a_2 X^2Z + a_4 XZ^2 + a_6 Z^3.$$

In affine coordinates this becomes

$$y^2 +a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.$$ (10.1)

Thus one often presents an elliptic curve by giving a Weierstrass equation (10.1.1), though there are significant computational advantages to other equations for curves (e.g., Edwards coordinates - see work of Bernstein and Lange).

Using Sage we plot an elliptic curve over the finite field $\mathbf{F}_7$ and an elliptic curve curve defined over $\mathbf {Q}$.

Note that both plots above are of the affine equation $y^2 = x^3 + x$, and do not include the distinguished point $\O$, which lies at infinity.