2.4. The group structure on E(Q) 25

P

Q

R

P+Q

Figure 1. Addition of points on an elliptic curve

E(Q). Let us find P + Q. First, we find the equation of the line

L = PQ. The slope must be

m =

0 − 6

5 − (−4)

= −

6

9

= −

2

3

and the line is L : y = −

2

3

(x − 5). Now we find the third point of

intersection of L and E by solving

y = −

2

3

(x − 5)

y2

=

x3

− 25x.

Plugging the first equation into the second one, we obtain an equation

x3

−

4

9

x2

−

185

9

x −

100

9

= 0,

which factors as (x − 5)(x + 4)(9x + 5) = 0. The first two factors

are expected, since we already knew that P = (5,0) and Q = (−4,6)

are in L ∩ E. The third point of intersection must have x = −

5

9

,

y = −

2

3

(x − 5) =

100

27

and, indeed, R = (−

5

9

,

100

27

) is a point in