aa r X i v : . [ m a t h . G M ] O c t A POINT OF ORDER 8
S. ADLAJ
Abstract.
A formula expressing a point of order 8 on an ellipticcurve, in terms of the roots of the associated cubic polynomial, isgiven. Doubling such a point yields a point of order 4 distinct fromthe well-known points of order 4 given in standard references suchas “A course of Modern Analysis” by Whittaker and Watson.
Let(1) y = 4( x − e )( x − e )( x − e ) , e i ∈ C , i = 1 , , , be the defining equation for an elliptic curve E over the complex field C , and let β := r e − e e − e , γ := p ( e − e )( e − e ) , i := √− . 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 thevalues β := s β + 1 β − r β − > β := r β + 1 + 1 β . The point P = ( x, y ) on E , where x = e − γ − γ r β + 12 − !(cid:18) − β + r β (cid:18)p β + β + i (cid:18)p β − β + r − β (cid:19)(cid:19)(cid:19) , is then a point of order 8.Note that doubly doubling the afore-indicated point P cannot possiblyyield either ( e ,
0) nor ( e , P is indeed of order 8,yield the point ( e , Date : May 30, 2011.1991
Mathematics Subject Classification.
Key words and phrases.
Torsion point on an elliptic curve over the complex field,roots of cubic, Weierstrass normal form, doubling formula.
S. ADLAJ
For an example, consider an elliptic curve E given in Weierstrass normalform via equation (1), where e = i, e = 0 , e = − i. Then γ = i √ , β = √ , β = 1+ √ r (cid:16) √ (cid:17) , β = 1 √ r (cid:16) √ − (cid:17) , and the x –coordinate of P is calculated to be x ( P ) = √ − − i r (cid:16) √ − (cid:17) . One might employ the well-known doubling formula, found in stan-dard sources such as [1], for successively calculating the points 2 P =(1 , ± √
2) and 4 P = (0 , § x -coordinates are e ± γ = i (1 ±√ P , as they match when doubled the points ( ± i, P , the two remaining points of order 2 on E. References [1] Whittaker E. T. Watson G. N. A Course of Modern Analysis. CambridgeUniversity Press; 4th edition (January 2, 1927).
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