Morton, Patrick2019-02-152019-02-152019Morton, P. (2019). Product formulas for the 5-division points on the Tate normal form and the Rogers-Ramanujan continued fraction. Journal of Number Theory. https://doi.org/10.1016/j.jnt.2018.12.013https://hdl.handle.net/1805/18391Explicit formulas are proved for the 5-torsion points on the Tate normal form E5 of an elliptic curve having (X,Y)=(0,0) as a point of order 5. These formulas express the coordinates of points in E5[5]−⟨(0,0)⟩ as products of linear fractional quantities in terms of 5-th roots of unity and a parameter u, where the parameter b which defines the curve E5 is given as b=(ε5u5−ε−5)/(u5+1) and ε=(−1+5–√)/2. If r(τ) is the Rogers-Ramanujan continued fraction and b=r5(τ), then the coordinates of points of order 5 in E5[5]−⟨(0,0)⟩ are shown to be products of linear fractional expressions in r(5τ) with coefficients in Q(ζ5).enPublisher PolicyTate normal form5-division pointsRogers–Ramanujan continued fractionArticle