70 2. Elliptic curves

δx + η. (Hint: multiply (2.14) by x and consider the change

of variables X = x and Y = xy. Make sure that, at the end,

the coeﬃcients of

y2

and

x3

equal 1.)

(3) Suppose that C /Q is a curve given by an equation of the

form

C :

y2

+ axy + by =

x3

+

cx2

+ dx + e. (2.15)

Find an invertible change of variables that takes the equa-

tion of C onto one of the form

y2

=

x3

+ Ax + B. (Hint:

do it in two steps. First eliminate the xy and y terms. Then

eliminate the

x2

term.)

(4) Let E/Q :

y2

+43xy−210y =

x3 −210x2.

Find an invertible

change of variables that takes the equation of E to one of

the form

y2

=

x3

+ Ax + B.

Exercise 2.12.4. Let C and E be curves defined, respectively, by

C : V 2 = U 4 + 1 and E : y2 = x3 − 4x. Let ψ be the map defined by

ψ(U, V ) =

2(V + 1)

U 2

,

4(V + 1)

U 3

.

(1) Show that if U = 0 and (U, V ) ∈ C(Q), then ψ(U, V ) ∈

E(Q).

(2) Find an inverse function for ψ; i.e., find ϕ : E → C such

that ϕ(ψ(U, V )) = (U, V ).

Next, we work in projective coordinates. Let C : W

2V 2

= U

4

+ W

4

and E :

zy2

=

x3

+

z3.

(3) Write down the definition of ψ in projective coordinates; i.e.,

what is ψ([U, V, W ])?

(4) Show that ψ([0, 1,1]) = [0,1,0] = O.

(5) Show that ψ([0, −1, 1]) = [0,0,1]. (Hint: Show that

ψ([U, V, W ]) = [2U

2,

4UW, W (V − W )].)

Exercise 2.12.5. Use Sage to solve the following problems:

(1) Find 3Q, where E :

y2

=

x3

− 25x and Q = (−4,6). Use

3Q to find a new right triangle with rational sides and area

equal to 5. (Hint: Examples 1.1.2 and 2.4.1.)