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.)
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