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Elementary Properties of the

Chord-Tangent Group Law on a Cubic Curve

In this chapter we illustrate how, by using simple analytic geometry, a large number
of numerical calculations are possible with the group law on a cubic curve. The cubic
3
2
2
curves in x and y will be in normal form, that is, without x y, xy ,or y terms. In
this form the entire curve lies in the affine x, y-plane with the exception of 0:0:1
which is to be zero in the group law. The lines through O are exactly the vertical
lines in the x, y-plane, and all other lines used are of the form y = λx + β.Weuse
the definition of P + Q as given in §5 of the Introduction.
We will postpone several questions related to the group law until Chapter 3 where
they are dealt with using results from Chapter 2 on the general theory of algebraic
curves. These include the associativity of the group law and a detailed discriminant
criterion for a curve to be nonsingular. The procedure for transforming a general
cubic into one in normal form will be worked out there too.

In Theorem (4.1) of this chapter there is a condition for an element (x , y ) on an

elliptic curve E(k) to be of the form 2(x, y) = (x , y ) in the group E(k). This plays

an important role in the proof of the Mordell theorem in Chapter 6.
In this chapter k will always denote a general field. For the beginning reader this
can for most considerations be viewed as the rational numbers Q.

§1. Chord-Tangent Computational Methods on a
Normal Cubic Curve

A cubic equation in normal form, or generalized Weierstrass form, is an expression
2 3 2
y + a 1 xy + a 3 y = x + a 2 x + a 4 x + a 6 ,
3
where the coefficients a i are in the field k. Since there is no term of the form y in
the equation, a vertical line x = x 0 intersects the locus of the normal cubic at two
points (x 0 , y 1 ) and (x 0 , y 2 ), where y 1 and y 2 are the roots of the quadratic equation
2
2
3
y +(a 1 x 0 +a 3 )y −(x +a 2 x +a 4 x 0 +a 6 ) = 0. In the completed plane, that is, the
0 0
projective plane, we see that the cubic in normal form has one additional solution at