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
