General form of Humbert's modular equation for curves with real multiplication of Δ = 5

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 ,...,x 6 the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to G 5 . We also prove the rationality of the hypersurface in P 5 defined by the generalized modular equation.