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

Kiichiro Hashimoto, Yukiko Sakai

    Research output: Contribution to journalArticle

    3 Citations (Scopus)


    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 x1;.. .; x6 the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to S{fraktur}5. We also prove the rationality of the hypersurface in P5 defined by the generalized modular equation.

    Original languageEnglish
    Pages (from-to)171-176
    Number of pages6
    JournalProceedings of the Japan Academy Series A: Mathematical Sciences
    Issue number10
    Publication statusPublished - 2009 Dec



    • Curves of genus two
    • Modular equation
    • Real multiplication

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this