ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 09 Jul 2017 06:20:18 -0500How to compute a cyclic subgroup of a class group?http://ask.sagemath.org/question/38198/how-to-compute-a-cyclic-subgroup-of-a-class-group/ Hello, I am quite new to computing in Sage and I am stuck at this point. Please help me out. I am going to use algebraic number theory notation without explaining them in detail. All computations are in quadratic fields.
What I have :
- $\Delta_p=p^2\Delta_K$ where $p$ is a prime, $\Delta_K=-pq$ and $\Delta_K\equiv 1$ mod $4$
- I computed the class group, $C(\Delta_p)$ in Sage
What I want :
- Set $f=[(p^2,p)]$ in $C(\Delta_p)$ where $(p^2,p)$ is the standard representation of an ideal of norm $p^2$
- Set $F=\langle f \rangle$
My question is :
- How to get that cyclic subgroup $F$?
I have been looking at AbelianGroup and ClassGroup in Sage but I am not quite understanding how to use these to actually get what I want.
Any help in this matter would be greatly appreciated.Sat, 08 Jul 2017 12:52:42 -0500http://ask.sagemath.org/question/38198/how-to-compute-a-cyclic-subgroup-of-a-class-group/Comment by dan_fulea for <p>Hello, I am quite new to computing in Sage and I am stuck at this point. Please help me out. I am going to use algebraic number theory notation without explaining them in detail. All computations are in quadratic fields.</p>
<p>What I have :</p>
<ul>
<li>$\Delta_p=p^2\Delta_K$ where $p$ is a prime, $\Delta_K=-pq$ and $\Delta_K\equiv 1$ mod $4$</li>
<li>I computed the class group, $C(\Delta_p)$ in Sage</li>
</ul>
<p>What I want :</p>
<ul>
<li>Set $f=[(p^2,p)]$ in $C(\Delta_p)$ where $(p^2,p)$ is the standard representation of an ideal of norm $p^2$</li>
<li>Set $F=\langle f \rangle$</li>
</ul>
<p>My question is :</p>
<ul>
<li>How to get that cyclic subgroup $F$?</li>
</ul>
<p>I have been looking at AbelianGroup and ClassGroup in Sage but I am not quite understanding how to use these to actually get what I want.</p>
<p>Any help in this matter would be greatly appreciated.</p>
http://ask.sagemath.org/question/38198/how-to-compute-a-cyclic-subgroup-of-a-class-group/?comment=38205#post-id-38205Let us see if your notation matches the guess of my sage and number theory impressions.
The following code initializes the imaginary quadratic number field of discriminant $-pq$, $p$ prime. (And $q$ too in my case.)
sage: p = 17
sage: q = 23
sage: DK = -p*q
sage: DK
-391
We initialize the quadratic field with the above discriminant and some order...
sage: K.<a> = QuadraticField( DK )
sage: K.discriminant()
-391
sage: Order = K.order( p*a )
sage: Order.discriminant().factor()
-1 * 2^2 * 17^3 * 23
sage: p
17
sage: Order.discriminant() == 4 * p^2 * DK
True
Well, we've got some factor four. If you do not expect it, please give us some explicit values for $p,q$.
Note: The class group is so far implemented only for the max order.Sun, 09 Jul 2017 06:20:18 -0500http://ask.sagemath.org/question/38198/how-to-compute-a-cyclic-subgroup-of-a-class-group/?comment=38205#post-id-38205Comment by tmonteil for <p>Hello, I am quite new to computing in Sage and I am stuck at this point. Please help me out. I am going to use algebraic number theory notation without explaining them in detail. All computations are in quadratic fields.</p>
<p>What I have :</p>
<ul>
<li>$\Delta_p=p^2\Delta_K$ where $p$ is a prime, $\Delta_K=-pq$ and $\Delta_K\equiv 1$ mod $4$</li>
<li>I computed the class group, $C(\Delta_p)$ in Sage</li>
</ul>
<p>What I want :</p>
<ul>
<li>Set $f=[(p^2,p)]$ in $C(\Delta_p)$ where $(p^2,p)$ is the standard representation of an ideal of norm $p^2$</li>
<li>Set $F=\langle f \rangle$</li>
</ul>
<p>My question is :</p>
<ul>
<li>How to get that cyclic subgroup $F$?</li>
</ul>
<p>I have been looking at AbelianGroup and ClassGroup in Sage but I am not quite understanding how to use these to actually get what I want.</p>
<p>Any help in this matter would be greatly appreciated.</p>
http://ask.sagemath.org/question/38198/how-to-compute-a-cyclic-subgroup-of-a-class-group/?comment=38203#post-id-38203Could you please provide the Sage code you already has ?Sun, 09 Jul 2017 03:30:08 -0500http://ask.sagemath.org/question/38198/how-to-compute-a-cyclic-subgroup-of-a-class-group/?comment=38203#post-id-38203