From: nikl+sm000463@pchelwig1.mathematik.tu-muenchen.de (Gerhard Niklasch)
Newsgroups: sci.math
Subject: Re: Imaginary Quadratic Fields with Class Number 2 or 3
Date: 28 Aug 1998 17:13:56 GMT
In article ,
jpr2718@aol.com (John Robertson) writes:
|> Does anyone have references giving the complete list of imaginary
|> quadratic fields with class number 2 and class number 3? Better yet,
|> is there a web site with these lists?
Partial answer:
h = 2 for discriminants
-15 | -20 | -24 | -35 | -40 | -51 | -52 | -88 | -91 | -115 | -123 |
-148 | -187 | -232 | -235 | -267 | -403 | -427
and for no others.
h = 3 for discriminants
-23 | -31 | -59 | -83 | -107 | -139 | -211 | -283 | -307 | -331 |
-379 | -499 ...
See
(the webmaster unfortunately(?) has long gotten distracted into
sharpening his tools instead of putting things into his number
field database).
One (perhaps not the best) paperspace reference is the Appendix
of J.-P. Serre, `Lectures on the Mordell-Weil Theorem', 2nd ed.,
Braunschweig: Vieweg 1990. There have been several recent journal
and proceedings articles about extending complete enumerations to
larger class numbers / harder class groups, but someone else may
be in a better position to provide references.
Enjoy, Gerhard
--
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From: hale@mailhost.tcs.tulane.edu (Bill Hale)
Newsgroups: sci.math
Subject: Re: Imaginary Quadratic Fields with Class Number 2 or 3
Date: Sat, 29 Aug 1998 01:19:08 -0500
In article , jpr2718@aol.com (John
Robertson) wrote:
> Does anyone have references giving the complete list of imaginary
> quadratic fields with class number 2 and class number 3? Better yet,
> is there a web site with these lists?
Look at the web page at the URL:
http://emmy.math.uni-sb.de/~georg/classne.html
It is a Java program that lists the (negatives of) discriminants of
imaginary quadratic fields that possess a given ideal class number h of
their maximal order.
For the reverse, look at my web page at the URL:
http://www.geocities.com/SiliconValley/Vista/3304/quadform.html
It is a Java program where you can enter a negative value for the
discriminant D and you will get the class number h(D) of the primitive
positive definite binary quadratic forms with discriminant D and a list
of all the reduced forms of discriminant D.
--
Bill Hale
==============================================================================
From: jpr2718@aol.com (Jpr2718)
Newsgroups: sci.math
Subject: Re: Imaginary Quadratic Fields with Class Number 2 or 3
Date: 29 Aug 1998 21:52:14 GMT
I wrote:
>Does anyone have references giving the complete list of imaginary
>quadratic fields with class number 2 and class number 3? Better yet,
>is there a web site with these lists?
Among other references, I have found the following:
Eric's Treasure Trove of Mathematics at
http://www.astro.virginia.edu/~eww6n/math/math.html under the entry for Class
Number lists discriminants with certain class numbers.
Sloane and Plouffe's book, The Encyclopedia of Integer Sequences, Academic
Press, 1995, gives the lists for class numbers 1, 2, and 3. The Class Number
entry in Eric's Treasure Trove of Mathematics, above, gives links to the
sequences in the online version of The Encyclopedia of Integer Sequences for
class numbers 1, 2, 3,4, 5, and 6.
Henri Cohen, A Course in Computational Algebraic Number Theory,
Springer-Verlag, page 229, gives complete lists for class numbers 1, 2, and 3.
==============================================================================
From: bobs@rsa.com
Newsgroups: sci.math
Subject: Re: Imaginary Quadratic Fields with Class Number 2 or 3
Date: Tue, 01 Sep 1998 15:09:20 GMT
In article <01bdd587$56dd0af0$2f26bf8a@wcwe00jr>,
"Rave Wolfgang" wrote:
> But actually I doubt there exists a _complete_ list for class numbers 2 and
> 3, as it took only 183 years to prove there are only 9 im. qu. fields with
> class number 1; but I'm not sure about it and I'm no professional
> mathematician but a number theory amateur.
What is it about Usenet that drives amateurs to post opinions even when
they are unsure of the facts?.
The first thing ANYONE should do, unless they KNOW, is to check the
existing literature.
Complete enumeration of all imaginary quadratic fields has been done
for h = 1,2,3,4,5,6,7. at least. The computations may have been extended.
There was a recent paper in Mathematics of Computation (around 1996,
I don't have a copy but remember reading the paper) which gave details
of the computations.
It is possible in principle to enumerate all fields for a given class number.
The computations can be prohibitive in some cases.
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From: elkies@abel.MATH.HARVARD.EDU (Noam Elkies)
Newsgroups: sci.math.numberthy
Subject: Re: Small Class Numbers
Date: 8 Apr 98 16:47:29 GMT
>Noam Elkies asked whether the list of Q(sqrt(-d)) with class number 4
>had been proved to be complete. This was done in 1992 by
>Steven Arno, Acta Arith 60 (1992), 321--334, MR 93b:11144.
>
>Has this been completed for h = 5, 6, ... ? Is h = 8 easier than
>h = 5?
Many respondents have already pointed out this Arno paper.
In general odd numbers are easier than evens, because the discriminant
must be prime. For instance, h=8 is the first case *not* handled
by any of the references brought to my attention.
Darrin Doud noted that Christian Wagner
has done 5,6,7 (Math. Comp. 65 (1996), pp 785-800); Doud and
Pieter Moree found the preprint
http://www.math.uiuc.edu/Algebraic-Number-Theory/0009/
by S.Arno, M.Robison and F.Wheeler (who also replied from
wheeler@super.org) that handles all odd class numbers through 23.
Wheeler suggests that 10 and 14 will be done before 8.
--Noam D. Elkies