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 RiemannRoch 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
20120924