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
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