Re: Computing generator of ideal I^n

[Jeroen Demeyer <jdemeyer@cage.ugent.be>]

On 08/07/2013 09:27 PM, Karim Belabas wrote:
> Sure! Unfortunately, it's awkward to do this in GP : the (libpari)
> function isprincipalfact() is used internally for this, see Libpari's
> manual, section "Computing in the class group"
>
>    (21:19) gp > ??"Computing in the class group"@
>    [...]
>
>     GEN isprincipalfact(GEN bnf,  GEN C,  GEN L,  GEN f, long flag) is about the
> same  as  bnfisprincipal0  applied  to  C prod L[i]^{f[i]},  where the L[i] are
> ideals, the f[i] integers and C is either an ideal or NULL (omitted). Make sure
> to include nf_GENMAT in flag!
>
>    ? install(isprincipalfact, GDGGGL)
>    ? K = bnfinit(y^2+23);
>    ? isprincipalfact(K,, K.gen, K.cyc,  4/*nf_GENMAT*/)
>    %2 = [[0]~, [[1, 1/2]~, 1; 2, 1]]
>
> The result is [e, t] as in bnfisprincipal(), with the important difference that
> 't' is given in *factored* form (which is undocumented in GP, but supported by
> most libpari number field functions).
>
> > It suffices to do this for the ideals generating the class group,
> > but that's already slow enough for the field that I'm using.
>
> In fact, this is often already part of the bnfinit structure (dynamically
> added as soon as a function requires it). It can be forced:
>
>    ? install(check_and_build_cycgen, G)
>    ? check_and_build_cycgen(K)
>    %3 = [Mat([[2, 1]~, 1])]
>
> (this is now part of K, all further calls will be instantaneous)
>
> This is a vector of generators for the g_i^d_i, where the g_i are the class
> group generators, and the d_i are their orders.

Thanks for this very interesting post!