The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 75

(b) Suppose t ( x 0 ) = 0. Use the facts that x 3 + Ax + B has no multiple
roots and all roots of t 2 are multiple roots to show that q ( x 0 )=0.
This shows that if q ( x 0 ) =0then α ( x 0 ,y 0 ) is defined.

2.20 Consider the singular curve y 2 = x 3 + ax 2 with a =0. Let y = mx
be a line through (0 , 0). Show that the line always intersects the curve
to order at least 2, and show that the order is 3 exactly w hen m 2 = a .

This may be interpreted as saying that the lines y = ± √ ax are the two
tangents to the curve at (0 , 0).

(a) Apply the method of Section 2.5.4 to the circle u 2 + v 2 2.21 =1and

the point ( − 1 , 0) to obtain the parameterization

u = 1 − t 2

2 t
1+ t 2 .

1+ t 2 ,

v =

(b) Suppose x, y, z are integers such that x 2 + y 2 = z 2 ,gcd( x, y, z )=1,
and x is even. Use (a) to show that there are integers m, n such
that

y = m 2 − n 2 , z = m 2 + n 2 .

x =2 mn,

Also, show that gcd( x, y, z ) = 1 implies that gcd( m, n )=1and
that m ≡ n (mod 2).

2.22 Let p ( x )and q ( x ) be polynomials with no common roots. Show that

p ( x )
q ( x )

=0

d
dx

(that is, the identically 0 rational function) if and only if both p ( x )=0
and q ( x )=0. (If p or q is nonconstant, then this can happen only in
positive characteristic.)

2.23 Let E be given by y 2 = x 3 + Ax + B over a field K and let d K × .The

twist of E by d is the elliptic curve E ( d ) given by y 2 = x 3 + Ad 2 x + Bd 3 .

∈

(a) Show that j ( E ( d ) )= j ( E ).
(b) Show that E ( d ) can be transformed into E over K ( √ d ).
(c) Show that E ( d ) can be transformed over K to the form dy 1 =

x 1 + Ax 1 + B .

2.24 Let α, β ∈ Z be such that gcd( α, β ) = 1. Assume that α ≡− 1(mod4)

0 (mod 32). Let E be given by y 2 = x ( x and β ≡ − α )( x − β ).

(a) Let p be prime. Show that the cubic polynomial x ( x − α )( x − β )
cannot have a triple root mod p .