2.10. Homogeneous spaces 63

(7) (δ1 = −1, δ2 = 2). The space C(−1, 2) has a rational point

(X, Y, Z) = (4,3,5), which maps to P = (−16, −120) on

E(Q) via Eq. (2.12). P is a point of infinite order.

(8) ((δ1, δ2) = (1,17), (34,1), or (−34,34)). These are the pairs

that correspond to (−1,2) · γ, with γ = (−1,34), (−34,2)

or (34,17). Therefore, the corresponding spaces C(δ1,δ2)

must have rational points that map to P + T1, P + T2 and

P + T1 + T2, respectively.

(9) (δ1 = −2, δ2 = 2). The space C(−2, 2) has a rational point

(X, Y, Z) = (1,4,3), which maps to Q = (−2, −48) on E(Q)

via Eq. (2.12). Q is a point of infinite order.

(10) ((δ1, δ2) = (2,17), (17,1), or (−17,34)). These are the pairs

that correspond to (−2,2) · γ, with γ = (−1,34), (−34,2)

or (34,17). Therefore, the corresponding spaces C(δ1,δ2)

must have rational points that map to Q + T1, Q + T2 and

Q + T1 + T2, respectively.

(11) ((δ1, δ2) = (2,1), and (−2,34), (−17,2), or (17,17)). Since

(−1,2) and (−2,2) correspond to P and Q, respectively,

then (−1,2) · (−2,2) = (2,1) corresponds to P + Q. The

other pairs correspond to (−2,2) · γ, with γ = (−1,34),

(−34,2) or (34,17). Therefore, the corresponding spaces

C(δ1,δ2) must have rational points that map to P + Q + T1,

P + Q + T2 and P + Q + T1 + T2, respectively.

(12) (δ1 = 1, δ2 = 2). The space C(1, 2) does not have ratio-

nal points (see Exercise 2.12.21). In fact, it does not have

solutions in Q2, the field of 2-adic numbers.

(13) ((δ1, δ2) = (2,2), (17,2), (34,2), (−1,1), (−2,1), (−17,1),

(−34,1), (−1,17), (−2,17), (−17,17), (−34,17), (1,34),

(2,34), (17,34), (34,34)). The corresponding spaces C(δ1,δ2)

do not have rational points. For instance, suppose C(2, 2)

had a point. Then (2,2,1) would be in the image of δ.

Since (2,1,2) is in the image of δ (we already saw above

that C(2, 1) has a point), then (2,1,2) · (2,2,1) = (1,2,2)

would also be in the image of δ, but we just saw (in the pre-

vious item) that (1,2,2) is not in the image of δ. Therefore,