Karim Belabas on Sat, 09 Jul 2005 20:47:04 +0200 |
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Re: Class groups |
* 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]