Product formulas for the $5$-division points on the Tate normal form and the Rogers-Ramanujan continued fraction

TL;DR

This paper derives explicit formulas for the 5-torsion points on a specific elliptic curve form, expressing their coordinates via products involving fifth roots of unity, a parameter, and the Rogers-Ramanujan continued fraction.

Contribution

It provides new explicit formulas for 5-torsion points on the Tate normal form using products of linear fractional quantities related to fifth roots of unity and the Rogers-Ramanujan continued fraction.

Findings

Formulas for 5-torsion points expressed in terms of roots of unity and parameter u.

Coordinates related to Rogers-Ramanujan continued fraction evaluated at 5τ.

Explicit algebraic expressions for points of order 5 on the elliptic curve.

Abstract

Explicit formulas are proved for the $5$-torsion points on the Tate normal form $E_5$ of an elliptic curve having $(X,Y)=(0,0)$ as a point of order $5$. These formulas express the coordinates of points in $E_5[5] - \langle(0,0)\rangle$ 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 $E_5$ is given as $b=(\varepsilon^5 u^5- \varepsilon^{-5})/(u^5+1)$ and $\varepsilon = (-1+\sqrt{5})/2$. If $r(\tau)$ is the Rogers-Ramanujan continued fraction and $b=r^5(\tau)$, then the coordinates of points of order $5$ in $E_5[5] - \langle(0,0)\rangle$ are shown to be products of linear fractional expressions in $r(5\tau)$ with coefficients in $\mathbb{Q}(\zeta_5)$.