  We describe explicit formulas relevant to the F-theory/heterotic string
duality that reconstruct from a specific Jacobian elliptic fibration on the
Shioda-Inose surface covering a generic Kummer surface the corresponding
genus-two curve using the level-two Satake coordinate functions. We derive
explicitly the rational map on the moduli space of genus-two curves realizing
the algebraic correspondence between a sextic curve and its Satake sextic. We
will prove that it is not the original sextic defining the genus-two curve, but
its corresponding Satake sextic which is manifest in the F-theory model, dual
to the $\mathfrak{so}(32)$ heterotic string with an unbroken
$\mathfrak{so}(28)\oplus \mathfrak{su}(2)$ gauge algebra.
