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

Determines the ray class field modulo ideal f over an
imaginary quadratic field.

<p><p>

<h3>
Syntax:
</h3>
<pre>O := ImQuadRayField( f [,"char"  |  "field"  |  "maxord"] );</pre>

<p>

<table>
<tr>
<td>ideal</td>
<td><pre>  f  </pre></td>
<td>integral ideal in imaginary quadratic field</td>
</tr>
<tr>
<td>order</td>
<td><pre>  O  </pre></td>
<td>ray class field modulo <tt> f</tt></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#RayClassField">RayClassField</a>, <a href="kashref.html#ImQuadHilbert">ImQuadHilbert</a>, <a href="kashref.html#OrderHilbert">OrderHilbert</a>

<p>

<h3>
Description:
</h3>
Let K be a imaginary quadratic field with maximal
order O|K, f an integral O|K-ideal
and K|{(f)} the ray class field over K corresponding to f.
Integral
generators for K|{(f)} over the Hilbert class field K|{(1)} are
for example elements of the form
 \Theta = \alpha\Big( P(1 |  f)-D \Big), 
where P(1 |  f) is a normalized singular value
of the Weierstrass \wp-function
and \alpha and D are suitable choosen elements
of K|{(1)} (see Scherz,Scherz2).
Furthermore we have an integral basis for K|{(f)}/K|{(1)}
given by
\begin{eqnarray*} O|{K|{(f)}} = O|{K|{(1)}} +
\ThetaO|{K|{(1)}} &+&
\alpha^{-1}\Theta^2O|{K|{(1)}} + &+&  &hellip;  +
\alpha^{-n+2}\Theta^{n-1}O|{K|{(1)}} \end{eqnarray*}
The minimal polynomial for \Theta is calculated using complex
approximations one gets by evaluating the defining infinite
series of the Dedekind \eta-function and the
Weierstrass \wp-function.

If <tt> "char"</tt> is specified only the characteristic polynomial
for \Theta over K is computed (which might be reducible).
In case of <tt> "field"</tt> also the Hilbert class field (and the
representation of the ray class field over
the Hilbert class field) is
determined, and if <tt> "maxord"</tt> or nothing is specified
(but that might cause a long output)
one gets the maximal order of the
ray class field as an extension of the maximal order of the Hilbert
class field.

<em> Note:</em> For numerical reasons don't use prime ideal powers
as a module whenever possible.
So for example if 2 is not inert in K, p|2 is a prime ideal
in K above 2 and f is prime to p|2, then we
have K|{(f)}=K|{(p|2f)} and one should
use <tt> ImQuadRayField</tt>(p|2f)
instead of <tt> ImQuadRayField</tt>(f).

<p><p>

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

<pre>

</pre>

<p>

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

Here Hilbert class field and imaginary field are the same (E1).
F[1] is the maximal order of the ray class field, compare also the
corresponding discriminants:
<pre>

</pre>

<p>


<hr>

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