The Satake sextic in elliptic fibrations on K3

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 $\mathfrak{s}\mathfrak{o}(32)$ heterotic string with an unbroken $\mathfrak{s}\mathfrak{o}(28)\oplus \mathfrak{s}\mathfrak{u}(2)$ gauge algebra.