2.9. Descent and the weak Mordell-Weil theorem 57 Thus, the image of δ is formed by the 16 elements that one obtains by multiplying out δ(T1), δ(T2), δ(P ) and δ(Q), in all possible ways. Thus, δ(E(Q)/2E(Q)) is the group: {(1, 1,1), (−7, −14, 2), (14,161,46), (−2, −46, 23), (−14, −7, 2), (2,2,1), (−1, −23, 23), (7,322,46), (2,1,2), (−14, −14, 1), (7,161,23), (−1, −46, 46), (−7, −7, 1), (1,2,2), (−2, −23, 46), (14,322,23)}. (Exercise: Check that the elements listed above form a group under multiplication.) We see that the only primes that appear in the fac- torization of the coordinates of elements in the image of δ are: 2,7 and 23. Therefore, the coordinates of δ are not just in Q×/(Q×)2 but in a much smaller subgroup of 16 elements: Γ = {±1, ±2, ±7, ±23, ±14, ±46, ±161, ±322} ⊂ Q×/(Q×)2. And the image of E(Q)/2E(Q) embeds into ΓΔ = {(δ1,δ2,δ3) ∈ Γ × Γ × Γ : δ1 · δ2 · δ3 = 1 ∈ Q×/(Q×)2} ⊂ Γ × Γ × Γ . Since Γ has 16 elements and E(Q)/2E(Q) embeds into (Γ )3, we conclude that E(Q)/2E(Q) has at most (16)3 = 212 elements. In fact, ΓΔ has only 162 elements, so E(Q)/2E(Q) has at most 28 elements. Notice also the following interesting “coincidence”: the prime divisors that appear in ΓΔ coincide with the prime divisors of the discriminant of E, which is ΔE = 6795034624 = 218·72·232. In the next proposition we explain that, in fact, this is always the case. Proposition 2.9.5. Let E : y2 = (x−e1)(x−e2)(x−e3), with ei ∈ Z. Let P = (x0,y0) ∈ E(Q) and write (x0 − e1) = au2, (x0 − e2) = bv2, (x0 − e3) = cw2, y0 2 = abc(uvw)2, where a, b, c, u, v, w ∈ Q, the numbers a, b, c ∈ Z are square-free, and abc is a square (in Z). Then, if p divides a · b · c, then p also divides the quantity Δ = (e1 − e2)(e2 − e3)(e1 − e3). Note: the discriminant of E equals ΔE = 16(e1 − e2)2(e2 − e3)2(e1 − e3)2. So a prime p divides Δ if and only if p divides ΔE. If p 2, then this is clear (see Exercise 2.12.19 for p = 2).

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