### Abstract

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 P^{5} defined by the generalized modular equation.

Original language | English |
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Pages (from-to) | 171-176 |

Number of pages | 6 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 85 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2009 Dec |

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### Keywords

- Curves of genus two
- Modular equation
- Real multiplication

### ASJC Scopus subject areas

- Mathematics(all)