Re: Class groups

[Karim Belabas <Karim.Belabas@math.u-psud.fr>]

* Mak Trifkovic [2005-07-05 05:21]:
> I have a quadratic extension K/F of arbitrary number fields, F with class 
> number one.  I have an order O in K (not necessarily maximal), and I would 
> like to have an explicit set of representatives for Pic(O), e.g. as rank 2 
> lattices over O_F.
> 
> Now I know PARI doesn't come anywhere near to having a command to do this, 
> so I've been trying to use various bnr commands.

See rnfsteinitz and rnfhnfbasis

> Some questions:
> 
> 1)  Is there documentation for the precise structure of the output of 
> bnrinit? 

No, sorry. It's rather painful to document and quite liable to change in
the near future [ just like nf, bnf and bid structures ]: most of the
information in there is not available in the most usable/efficient form
( a number of components are already obsoleted ).

> Does it actually initialize the class field of a given conductor, 
> or at least give me the relative polynomial?

No. It's much easier to compute a full class field theoretic description
than a defining polynomial.

You can use 
-- bnrstark (for totally real class fields), or
-- rnfkummer (extension of _prime_ degree).

It's in general faster to compute rnfkummer(bnr,,p) rather than
rnfkummer(bnr,subgroup) for some subgroup of index p. The former computes
in one shot _all_ extensions of prime degree p with the same conductor
(conductor, not modulus!) as the ray class field associated to the bnr. An
arbitrary extension is obtained by repeatedly using this method.

Cheers,

    Karim.
-- 
Karim Belabas                     Tel: (+33) (0)1 69 15 57 48
Dep. de Mathematiques, Bat. 425   Fax: (+33) (0)1 69 15 60 19
Universite Paris-Sud              http://www.math.u-psud.fr/~belabas/
F-91405 Orsay (France)            http://pari.math.u-bordeaux.fr/  [PARI/GP]