# Groups Attached to Elliptic Curves

Definition 10.1.1 (Elliptic Curve)   An elliptic curve over a field  is a genus one curve  defined over  equipped with a distinguished point .

We will not define genus in this book, except to note that a nonsingular curve over  has genus one if and only if over  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

In affine coordinates this becomes

 (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 and an elliptic curve curve defined over .

Note that both plots above are of the affine equation , and do not include the distinguished point , which lies at infinity.

Subsections
William Stein 2012-09-24