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§5. An Explicit 2-Isogeny 97

2

b y 2 4 2 2
2
2
(y ) = y 1 − = x − 2x b + b ,
x 2 x 4
x 6 2 6 4 2 2
(y ) = x − 2bx + b x ,
y 2
and
y 6 y 4 y 2
2
f (x ) = 6 − 2a 4 + a − 4b 2 ,
x x x
x 6 4 2 2 2 4

f (x ) = y − 2ay x + a − 4b x ,
y 2
3
6
2
2
2
2
which boils down to (x /y )(y ) using y = x + ax + bx.
(5.3) Remark. Next we calculate ˆϕϕ as follows:

2
y 2 y(x − b)
ˆ ϕ(ϕ(x, y)) =ˆϕ ,
x 2 x 2
2 2 2 4 2 4 4
y (x − b) x y(x − b) x y 2
= · , · − (a − 4b)
4x 4 y 4 8x 2 y 4 x 4
2 2 2
(x − b) x − b 4 2 4
= , · y − a − 4b x .
2 3
4y 2 8x y
In order to show that ˆϕ(ϕ(x, y)) = 2(x, y), we consider the tangent line to a point
(x, y) on the curve. To find the slope of the tangent line, we differentiate the equation
for E[a, b] and we obtain
2
2
2

2yy = 3x + 2ax + b = 2(x + ax + b) + (x − b)
2y 2 2
= + (x − b),
x
or, in other words,
2
dy y x − b
= + .
dx x 2y
If2(x 1 , y 1 ) = (x 2 , y 2 ) on E[a, b], then the tangent line y = σ(x − x 1 ) + y 1 to
E[a, b]at (x 1 , y 1 ) must intersect E[a, b]at (x 2 , −y 2 ). In particular, x 2 is a root and
x 1 is a double root of the cubic equation
3
2
x + ax + bx = (σ(x − x 1 ) + y 1 ) 2
or

3
2
0 = x + a − σ 2 x +· · ·
from which we deduce that