N.Bruin on Wed, 9 Sep 1998 16:42:29 +0200 (MET DST)

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 Number field with prescribed integral basis

```Dear recipient,

I would like to have a representation of a number field K (a bnf,
say) with a prescribed integral basis. I want the basis to be LLL reduced,
to be precise. The reason is, that I want to produce small elements that
satisfy certain congruency-relations, i.e

v=v_0+u^t*v_1 mod P^e,

where u is a uniformizer of the prime ideal P, v_0 is a prescribed element
(mod P^t) and v_1 loops through representatives of Z_K / P^(e-t).

I want v to be small in some sense (I don't know the "best" sense for my
situation. I just want to generate potential x-coordinates for K-rational
points on al elliptic curve)

One way I hope to achieve this, is make sure v has small cooordinates wrt.
an LLL-reduced basis. An nfeltreduce mod P^e would do that, if the nf
given has an LLL-reduced basis.

In KASH this is quite easily achieved and gave me some results. However,
now I want to write a special program to obtain maximum speed and Pari/GP
seems the way to go.

Does anyone know how to get pari to work wrt. a prescribed basis? What I'm
searching for is something that updates the other entries in the nf
datastructure, given an integral basis.

Also, other suggestions for the problem stated are quite welcome.
(the problem is that I want to find Mordell-Weil generators of an elliptic
curve over a 12-th degree extension of Q, so computational work gets out
of hand pretty quickly)

Thanks for your attention,

Nils Bruin
nbruin@wi.leidenuniv.nl
```