- --- In primenumbers@yahoogroups.com, mikeoakes2@a... wrote:

> And h(-44) = 3 - see e.g. H.Cohen "A Course in Computational

I've found two places on the web where I think h(-44)=3 ought to show

>Algebraic Number Theory" (Springer, 1996), Appendix B.

up, and it isn't in either place:

http://mathworld.wolfram.com/ClassNumber.html

http://www.research.att.com/cgi-bin/access.cgi/as/

njas/sequences/eisA.cgi?Anum=A006203

I also see some differences from your list here:

> h(-4*n) = 1 for n=1,2,3,7.

12 and 28 don't show up for h(-d)=1.

> h(-4*n) = 2 for n=5,6,10,13,15,22,37,58.

60 doesn't show up for h(-d)=2.

Am I doing something wrong, or so these sources disagree with your

source? - In a message dated 07/07/03 04:23:08 GMT Daylight Time, sleephound@...

writes:

> I also see some differences from your list here:

h(-12) = h(-28) = 1 give n = 3 resp. 7.

>

> > h(-4*n) = 1 for n=1,2,3,7.

> > h(-4*n) = 2 for n=5,6,10,13,15,22,37,58.

>

> 12 and 28 don't show up for h(-d)=1.

> 60 doesn't show up for h(-d)=2.

>

> Am I doing something wrong, or so these sources disagree with your

> source?

>

h(-60) = 2 gives n = 15.

Ok?

Notice that I said "..(squarefree) n...". The list for /all/ n is a bit

longer:-

h(-4*n) = 1 for n=1,2,3,4,7.

h(-4*n) = 2 for n=5,6,8,9,10,12,13,15,16,18,22,25,28,37,58.

In fact, there's quite a serious error in my earlier claim that> it is necessary that the class number h(-4*n) = 1 or 2, since that is the

Further reading has made it clear that not only the great Gauss (1801) but

> degree of the polynomial in the subsidiary condition

even before him the mighty Euler (c. 1750) were in posession of /65/ values of n

which have a "linear" characterisation of the type we are discussing.

To the above must be added (using our modern notation of class numbers of

fields):-

h(-4*n) = 4 for n=21,24,30,33,40,42,45,48,57,60,70,72,

78,85,88,93,102,112,130,133,177,190,232,253

h(-4*n) = 8 for n=105,120,165,168,210,240,273,280,312,330,

345,357,385,408,462,520,760.

h(-4*n) = 16 for n=840,1320,1365,1848.

[These values come from David Cox's superb book.]

Apparently it has since been proved that there is at most one other value of

n, but no-one has been able to show whether or not it exists!

This is one of the deepest and richest areas of number theory, still not

exhausted by centuries of mining.

Mike Oakes

[Non-text portions of this message have been removed] - In a message dated 07/07/03 02:08:44 GMT Daylight Time,

tomabeti@... writes:

> Hilbert polynomial of the quadratic field with discriminant -44 is

Thanks!

> x^3-1122662608*x^2+270413882112*x-653249011576832

>

Can you reduce it? (I'm no pari expert) - it ought to be equivalent to my

cubic.

Mike Oakes

[Non-text portions of this message have been removed] - On Mon, 7 Jul 2003 04:31:45 EDT

mikeoakes2@... wrote:

> > Hilbert polynomial of the quadratic field with discriminant -44 is

x^3 - x^2 + x + 1

> > x^3-1122662608*x^2+270413882112*x-653249011576832

> >

>

> Thanks!

> Can you reduce it? (I'm no pari expert) - it ought to be equivalent to my

> cubic.

David Broadhurst told me:>? polred(x^3-1122662608*x^2+270413882112*x-653249011576832)

It says that two polynomials define the same (class) filed.

>%1 = [x - 1, x^3 - x^2 - x - 1, x^3 - x^2 + x + 1]

>so x^3-x^2-x+1 seems to do the same job?

You can use another Weber polynomial x^3 - 2*x^2 + 2*x - 2,

which also defines the same field.

Satoshi Tomabechi - In a message dated 08/07/03 02:14:47 GMT Daylight Time,

tomabeti@... writes:

>> Can you reduce it? (I'm no pari expert) - it ought to be equivalent to my

Thank you.

>> cubic.

>

> x^3 - x^2 + x + 1

Meanwhile, (pari Grand Master) David Broadhursthas emailed me with> henri cohen's "minimal height" form for your d=-44 cubic is

So that's good ! - there's complete agreement between your theoretical

> x^3-x^2+x+1

Hilbert polynomial and my experimentally discovered cubic.

Mike Oakes

[Non-text portions of this message have been removed]