2.2. Definition 21
but with the understanding that in this new model we may have left
out some points of E at infinity (i.e., those points [X, Y, 0] satisfying
In general, one can find a change of coordinates that simplifies
Eq. 2.3 enormously:
Proposition 2.2.2. Let E be an elliptic curve, given by Eq. 2.2,
defined over a field K of characteristic different from 2 or 3. Then
there exists a curve E given by
A, B ∈ K with
and an invertible change of variables ψ : E → E of the form
ψ([X, Y, Z]) =
f1(X, Y, Z)
g1(X, Y, Z)
f2(X, Y, Z)
g2(X, Y, Z)
f3(X, Y, Z)
g3(X, Y, Z)
where fi and gi are polynomials with coeﬃcients in K for i = 1,2,3,
and the origin O is sent to the point [0,1,0] of E, i.e., ψ(O) = [0,1,0].
The existence of such a change of variables is a consequence of
the Riemann-Roch theorem of algebraic geometry (for a proof of the
proposition see [Sil86], Chapter III.3). The reference [SiT92], Ch. I.
3, gives an explicit method to find the change of variables ψ : E → E.
See also pages 46-49 of [Mil06].
A projective equation of the form
in aﬃne coordinates, is called a Weierstrass equation.
From now on, we will often work with an elliptic curve in this form.
Notice that a curve E given by a Weierstrass equation y2 = x3 +Ax+
B is non-singular if and only if 4A3 + 27B2 = 0, and it has a unique
point at infinity, namely [0,1,0], which we shall call the origin O or
the point at infinity of E.
Sometimes we shall use a more general Weierstrass equation
+ a1xy + a3y =
+ a4x + a6
with ai ∈ Q (we will explain the funky choice of notation for the
coeﬃcients later), but most of the time we will work with equations
of the form
+ Ax + B. It is easy to come up with a change
of variables from one form to the other (see Exercise 2.12.3).