Could you please tell me what is the points of order 3 (or 6) Hon the elliptic curve $y^2=x^3+sx^2x$ where $$s = \frac{1}{432}\frac{(81k^82592k^46912)}{k^6}$$ and $k$ is rational?
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Two points of order $6$ are $(4/k^2, \pm (k^4+16)/k^5)$, and two points of order $3$ are $(k^2/4, \pm (1+k^4/16)/k)$. This can be verified via the following Magma code, which can be used on the free Magma online calculator:
which outputs


