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We show how to calculate the zeta functions and the orders |X| of Tate-Shafarevich groups of the elliptic curves with equation Y 2 + XY = X 3 + αX 2 + const · T −k over the rational function field Fq(T), where q is a power of 2. In the range q = 2, k ≤ 37, α ∈ F 2 [T −1 ] odd of degree ≤ 19, the largest values obtained for |X| are 47 2 (one case), 39 2 (one… (More)

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