We study Humbert's modular equation which characterizes curves of genus two having real multiplication by the quadratic order of discriminant 5. We give it a simple, but general expression as a polynomial in $x_1,\ldots ,x_6$ the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to $\frak S_5$. We also prove the rationality of the hypersurface in P5 defined by the generalized modular equation.
"General form of Humbert's modular equation for curves with real multiplication of $\Delta =5$." Proc. Japan Acad. Ser. A Math. Sci. 85 (10) 171 - 176, December 2009. https://doi.org/10.3792/pjaa.85.171