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

$\displaystyle 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

$\displaystyle y^2 +a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.$ (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}$.
sage: E = EllipticCurve(GF(7), [1,0])
sage: E.plot(pointsize=50, gridlines=True)

sage: E = EllipticCurve([1,0])
sage: E.plot()

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.

William Stein 2012-09-24