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Miura :: Miura

Miura -- A Package for Miura Curve Arithmetic


The Miura package realizes arithmetic on the curves such as hyper-elliptic curves (e.g., y^2=x^5+x+1), C_{ab} curves (e.g., y^3=x^4+2x+1), complete intersection (e.g. {y^2-x^3-1,z^2-x*y-1}). For the Miura form, the pole orders should be specified such as 2 and 3 for x and y of an elliptic curve. Currently, only divisor class group computation is available for the package. For the elliptic curves, [(P)-(O)]+[(Q)-(O)] = [(P+Q)-(O)] for two points P, Q and the point O at infinity. For the general nonsingular curves, any divisor class is uniquely expressed by E-g(O) with E a positive divisor of degree g (genus). This package reduces the divisor class group addition to ideal class group multiplication, and utilizes Groebner basis computation. See for the detail



This documentation describes version 0.2 of Miura.

Source code

The source code from which this documentation is derived is in the file Miura.m2.


  • Functions and commands
    • add -- Add Reduced Ideals
    • double -- Double Reduced Ideal
    • reduced -- Compute Reduced Ideal
    • scalarMultiplication -- Add Reduced Ideal Multiple Times
    • setPolynomialRing (missing documentation)
    • setQuotientRing (missing documentation)
  • Symbols
    • PR (missing documentation)
    • QR (missing documentation)