  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
$\mathbb{Q}(\zeta_5)$.
