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<h2><a name="AlffClassGroup"></a>
AlffClassGroup
</h2>

Computes the structure of the divisor class group of a
global function field.

<p><p>

<h3>
Syntax:
</h3>
<pre>Cl := AlffClassGroup( F [, b ] [, A ] [, "fast" ] );</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  Cl  </pre></td>
<td>of integer and list (class group structure)</td>
</tr>
<tr>
<td>alff</td>
<td><pre>  F  </pre></td>
<td>global function field</td>
</tr>
<tr>
<td>integer</td>
<td><pre>  b  </pre></td>
<td>bound for degree of places to be used</td>
</tr>
<tr>
<td>alff divisor</td>
<td><pre>  A  </pre></td>
<td>for reduction</td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#AlffClassGroupGenBound">AlffClassGroupGenBound</a>, <a href="kashref.html#AlffClassNumberApprox">AlffClassNumberApprox</a>, <a href="kashref.html#AlffDivisorDegOne">AlffDivisorDegOne</a>, <a href="kashref.html#AlffLPoly">AlffLPoly</a>, <a href="kashref.html#Alff">Alff</a>, <a href="kashref.html#AlffInit">AlffInit</a>, <a href="kashref.html#AlffOrders">AlffOrders</a>

<p>

<h3>
Description:
</h3>
Let F/k be a global function field. The divisor class group
is the factor group of all divisors factored by the principle
divisors.
This function returns a list <tt> Cl</tt>:
<tt> Cl[1]</tt> is the class number and <tt> Cl[2]</tt> is
a list of integers 1 < c_1 | \dots | c_m
such that the class group
of F/k is isomorphic to Z/c_1Z \times \dots \times Z/c_mZ.
The function requires that the constant field of definition is
the exact constant field.
\medskip

The following additional parameters can be given:
<ul>
<li> b is a bound for the places to be considered in the
factor base. For large q or g it can be reasonable to take
smaller values for b than returned by
<tt> AlffClassGroupGenBoundStrong(F)</tt>.
<li> A is a reduction divisor. It must factor over the
factor base (which is tested automatically)
and should have small, positive degree. If the
degree [F : k(T)] is large than it can be reasonable to take
e.g. <tt> A1 := AlffDivisorDegOne(F)</tt>.
<li> If b is chosen very small and if the class group
is a product of many cyclic factors then the option <tt> "fast"</tt>
gives a speedup. The result of the computation however is
not proven be correct anymore.
</ul>
The algorithm is described in He2.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

</pre>

<p>


<hr>

<p>
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