A point of order 8

arXiv:1110.0357v1 [math.GM] 3 Oct 2011 A POINT OF ORDER 8 S. ADLAJ Abstract. A formula expressing a point of order 8 on an elliptic curve, in terms of the roots of the associated cubic polynomial, is given. Doubling such a point yields a point of order 4 distinct from the well-known points of order 4 given in standard references such as “A course of Modern Analysis” by Whittaker and Watson. Let (1) y2 = 4(x −e1)(x −e2)(x −e3), ei ∈C, i = 1, 2, 3, be the defining equation for an elliptic curve E over the complex field C, and let β := re1 −e3 e1 −e2 , γ := p (e1 −e3)(e1 −e2), i := √ −1. The roots of the cubic on the right hand side of the defining equation (1) need not sum to zero but assume that β > 1, and introduce the values β1 := s β + 1 β −1 + r 2 β −1 > β2 := r 2 β + 1 + 1 β . The point P = (x, y) on E, where x = e1 −γ −γ r β + 1 2 −1 !  1 −1 β + r 1 + 1 β p β1 + β2 + i p β1 −β2 + r 1 −1 β  , is then a point of order 8. Note that doubly doubling the afore-indicated point P cannot possibly yield either (e1, 0) nor (e3, 0), and so must, if P is indeed of order 8, yield the point (e2, 0). Date: May 30, 2011. 1991 Mathematics Subject Classification. 11G05. Key words and phrases. Torsion point on an elliptic curve over the complex field, roots of cubic, Weierstrass normal form, doubling formula. 2 S. ADLAJ For an example, consider an elliptic curve E given in Weierstrass normal form via equation (1), where e1 = i, e2 = 0, e3 = −i. Then γ = i √ 2, β = √ 2, β1 = 1+ √ 2+ r 2 √ 2 + 1  , β2 = 1 √ 2+ r 2 √ 2 −1  , and the x–coordinate of P is calculated to be x(P) = √ 2 −1 −i r 2 √ 2 −1  . One might employ the well-known doubling formula, found in stan- dard sources such as [1], for successively calculating the points 2P = (1, ±2 √ 2) and 4P = (0, 0), the latter evidently being a point of order 2 on E. Incidentally, the formulas given in [1, §20.33, p. 444] apply here yielding two pairs of points of order 4, whose x-coordinates are e1 ± γ = i(1 ± √ 2), so all four points differ from either point of the already computed pair 2P, as they match when doubled the points (±i, 0), which are, aside from the point 4P, the two remaining points of order 2 on E. References [1] Whittaker E. T. Watson G. N. A Course of Modern Analysis. Cambridge University Press; 4th edition (January 2, 1927). CC RAS, Vavilov st. 40, Moscow, Russia, 119333 E-mail address: SemjonAdlaj@gmail.com

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