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 2 V 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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