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<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="OrderReg"></a>
OrderReg
</h2>

Returns the regulator of the <em> current</em>
maximal system of independent units.

<p><p>

<h3>
Syntax:
</h3>
<pre>x := OrderReg(o);
x := OrderReg(o,"classgroup");
x := OrderReg(o, reg);</pre>

<p>

<table>
<tr>
<td>real</td>
<td><pre>  x  </pre></td>
<td></td>
</tr>
<tr>
<td>order</td>
<td><pre>  o  </pre></td>
<td></td>
</tr>
<tr>
<td>real</td>
<td><pre>  reg  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#OrderRegLowBound">OrderRegLowBound</a>, <a href="kashref.html#OrderUnitsFund">OrderUnitsFund</a>, <a href="kashref.html#OrderUnitsIndep">OrderUnitsIndep</a>

<p>

<h3>
Description:
</h3>
Let o be an absolute order with unit rank r \ge 1.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Let O denote the maximal order of Q(rho), where
rho^4 + 73 rho^2 - 280 rho - 2399 = 0. First, we
compute the regulator of a maximal system of independent
units in O. 
<pre>

kash> O := OrderMaximal (Poly(Zx,[1,0,73,-280,-2399]));
   F[1]
    |
   F[2]
  /
 /
Q
F  [ 1]     Given by transformation matrix
F  [ 2]     x^4 + 73*x^2 - 280*x - 2399
Discriminant: -997975 

kash> OrderUnitsIndep (O);
[ [0, 0, 1, -1], [109589, -13889, 743946, -766419] ]
kash> OrderReg (O); 
12.399824579954190488548880525089080810892337788058

</pre>

<p>

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


Next we compute the regulator of a fundamental
system of units in O. 
<pre>

kash> OrderUnitsFund (O);
[ [-1, 0, -1, 1], [-253011, -13889, 519847, -542320] ]
kash> OrderReg (O); 
12.39982457995419048854888052508908081089233778807

</pre>

<p>

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


From this result we find that the system
of independent units which we had computed in the first step
was fundamental. \bigskip

Finally, we set the regulator of the maximal order to 12.4:
<pre>

kash> OrderReg (O, 12.4);
12.4
kash> OrderReg (O);
> 12.4

</pre>

<p>


<hr>

<p>
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