64 2. Elliptic curves we have reached a contradiction and C(2, 2) cannot have a rational point. One can rule out all the other (δ1,δ2) in the list similarly. We have analyzed all 64 possible pairs (δ1,δ2) and have found that the image of E(Q)/2E(Q) via δ has order 24. Therefore, 2RE+2 = 24 and RE = 2. The rank of the curve is exactly 2 and T1,T2,P and Q (as found above) are generators of E(Q)/2E(Q). (In fact, they are generators of E(Q) as well.) Example 2.10.5. Let E : y2 = x3 − 6724x = x(x − 82)(x +82). Let e1 = 0, e2 = −82 and e3 = 82. The only divisors of ΔE are 2 and 41, hence Γ = {±1, ±2, ±41, ±82}. Let us analyze the homogeneous spaces C(δ1,δ2) : δ2Y 2 − δ1X2 = 82, δ2Y 2 − δ1δ2Z2 = 164 as we did in the previous example. Once again, there are 64 pairs to check: (1) ((δ1, δ2,δ3) = (1,1,1)). The point at infinity (i.e., the origin) is sent to (1,1,1) via δ, i.e., δ(O) = (1,1,1). (2) (δ1 0 and δ2 0). The equation δ2Y 2 − δ1δ2Z2 = 164 cannot have rational solutions because the left-hand side is always negative for any X, Z ∈ Q. (3) (δ1 0 and δ2 0). The equation δ2Y 2 −δ1X2 = 82 cannot have rational solutions, because the left-hand side is always negative. (4) ((δ1, δ2) = (−1,82), (−82,2), (82,41)). The corresponding spaces have (trivial) rational points that map, respectively, to T1 = (0,0), T2 = (−82,0) and T3 = T1 + T2 = (82,0) via Eq. (2.12). (5) ((δ1, δ2) = (1,2)). The space C(1, 2) does not have rational points (same reason as for Exercise 2.12.21). In fact, it does not have any solutions over Q2. (6) ((δ1, δ2) = (−1,41), (−82,1), (82,82)). The correspond- ing spaces cannot have rational points, because these ele- ments of ΓΔ are the product of (1,2,2), with no points,

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