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    <title>Miura -- Miura curve arithmetic</title>
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<a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Miura curve arithmetic" href="index.html">Miura</a> :: <a title="Miura curve arithmetic" href="index.html">Miura</a></span>      </div>
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      <h1>Miura -- Miura curve arithmetic</h1>
      <div>
        <h2>Description</h2>
        <p>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 http://arxiv.org/pdf/1512.08040v1.pdf for the detail</p>
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      <div>
        <div>
          <div>
            <h2>Author</h2>
            <ul>
              <li><a href="http://suzuki.bayesnet.org">Joe Suzuki</a><span> &lt;<a href="mailto:j-suzuki%40sigmath.es.osaka-u.ac.jp">j-suzuki@sigmath.es.osaka-u.ac.jp</a>></span></li>
            </ul>
          </div>
          <div>
            <h2>Version</h2>
            <p>This documentation describes version <b>0.2</b> of Miura, released <b>5 October 2017</b>.</p>
          </div>
          <div>
            <h2>Citation</h2>
            <p>If you have used this package in your research, please cite it as follows:</p>
            <table class="examples">
              <tr>
                <td>
                  <pre><code class="language-bib">@misc{MiuraSource,
  title = {{Miura: arithmetic of Miura curves. Version~0.2}},
  author = {Joe Suzuki},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}
</code></pre>
                </td>
              </tr>
            </table>
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          <div>
            <h2>Exports</h2>
            <div class="exports">
              <ul>
                <li>Functions and commands                  <ul>
                    <li><span><a title="Add Reduced Ideals" href="_add.html">add</a> -- Add Reduced Ideals</span></li>
                    <li><span><a title="Double Reduced Ideal" href="_double.html">double</a> -- Double Reduced Ideal</span></li>
                    <li><span><a title="Compute Reduced Ideal" href="_reduced.html">reduced</a> -- Compute Reduced Ideal</span></li>
                    <li><span><a title="Add Reduced Ideal Multiple Times" href="_scalar__Multiplication.html">scalarMultiplication</a> -- Add Reduced Ideal Multiple Times</span></li>
                    <li><span><span class="tt">setPolynomialRing</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">setQuotientRing</span> (missing documentation)
</span></li>
                  </ul>
                </li>
                <li>Methods                  <ul>
                    <li><span><span class="tt">add(Ideal,Ideal)</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">double(Ideal)</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">reduced(Ideal)</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">scalarMultiplication(Ideal,ZZ)</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">setPolynomialRing(GaloisField,List,List)</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">setPolynomialRing(List,List)</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">setQuotientRing(Ideal)</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">setQuotientRing(List)</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">setQuotientRing(RingElement)</span> (missing documentation)
</span></li>
                  </ul>
                </li>
                <li>Symbols                  <ul>
                    <li><span><span class="tt">PR</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">QR</span> (missing documentation)
</span></li>
                  </ul>
                </li>
                <li>Mutable symbols                  <ul>
                    <li><span><span class="tt">PR</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">QR</span> (missing documentation)
</span></li>
                  </ul>
                </li>
              </ul>
            </div>
          </div>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="Miura curve arithmetic" href="index.html">Miura</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Miura.m2</span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">Miura.m2:90:0</span>.</p>
        </div>
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