# Question: suppose that p x y is a point on...

###### Question details

Suppose that P = (x, y) is a point on the cubic curve

y2 = x3 + ax2 + bx + c.

(a) Verify that the x-coordinate of the point 2P is given by the duplication formula

x(2P) =

x4 − 2bx2 − 8cx + b2 − 4ac

4x3 + 4ax2 + 4bx + 4c

.

(b) Derive a similar formula for the y-coordinate of 2P in terms of x and y.

(c) Find a polynomial in x whose roots are the x-coordinates of the point P =

(x, y) satisfying 3P = O. (Hint. The relation 3P = O can also be written as

2P = −P.)

(d) For the particular curve y2 = x3 + 1, solve the equation in (c) to find all points

satisfying 3P = O. Note that you will need to use complex numbers.