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26 1. Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve

y 1 − y 2
λ = .
x 1 − x 2

Case 2. If x 1 = x 2 but P 1 = P 2 , then the line through P 1 and P 2 is the vertical
line x = x 1 and P 2 =−P 1 as in (1.2).
Case 3. If P 1 = P 2 , then the tangent line to P 1 has the equation y = λx + β,
where

f (x 1 ) − a 1 y 1
λ =
2y 1 + a 1 x 1 + a 3

since (2y + a 1 x 1 + a 3 )y = f (x) − a 1 y after differentiating implicitly the normal

form of the cubic equation.
Now substituting y = λx + β into the normal form of the cubic equation, and
collecting all terms on one side, we have the following relations:
2 3 2
(λx + β) + a 1 x(λx + β) + a 3 (λx + β) = x + a 2 x + a 4 x + a 6
and

3
2
2
2
0 = x + (a 2 − λ − λa 1 )x + (a 4 − 2λβ − a 1 β − λa 3 )x + (a 6 − β − a 3 β).
The three roots of this cubic equation are x 1 , x 2 ,and x 3 ,the x-coordinates of the
three intersection points, either P 1 , P 2 ,and P 1 P 2 in Case 1 or P 1 , P 1 ,and P 1 P 1 in
2
Case 3. Since the sum of the roots is the negative of the coefficient of x in the cubic
equation for x, we have the following formula for x 3 :
2
x 3 = λ + λa 1 − a 2 − x 1 − x 2 for Case 1, Case 2
2
= λ + λa 1 − a 2 − 2x 1 for Case 3,
and the y-coordinate is given by the equation of the line

y 3 = λx 3 + β.
Finally,

(x 1 , y 1 ) + (x 2 , y 2 ) = (x 3 , −y 3 − a 1 x 3 − a 3 ) for Case 1, Case 2

and
2(x 1 , y 1 ) = (x 3 , −y 3 − a 1 x 3 − a 3 ) for Case 3.

3
2
(1.5) Example. Return to the elliptic curve E: y + y − xy = x of Example (1.3).
Denote by P the point (1, 1) on E, and observe that the tangent line T to P cuts
thecubicat (0, −1) =−(0, 0). Thus 2P = (0, 0) by the procedure in (1.4). Next
observe that y = x is the line through P = (1, 1) and 2P = (0, 0), and the third
point of intersection is (−1, −1) =−(−1, −1).Hence 3P = (−1, −1). Further