Generator matrix of ideal lattice

[Cindy]

Hi,

Let K be a number field and O_k denote its ring of integers. For an ideal, J of O_k, we can have an ideal lattice (I,b_\alpha), where

b_\alpha: J\times J \to Z, b_\alpha(x,y)=Tr(\alpha xy), \forall x,y \in J

and \alpha is a totally positive element of K\{0}.

Suppose now I know J and \alpha, how can I get the generator matrix for the ideal lattice (J,\alpha) using sage?

Thanks a lot.

Cindy

[David Loeffler]

The first thing I tried was this, and it seems to work fine:

sage: K.<z> = NumberField(x^3 - x + 17)
sage: I = K.primes_above(17)[1]
sage: alpha = 13*z + 4
sage: matrix([[(u*v*alpha).trace() for u in I.basis()] for v in I.basis()])
[ 3468 646 -11339]
[ 646 -591 -871]
[-11339 -871 225]

David

[Cindy]

Hi, David,

Thanks a lot! It works.^^

Cindy

[Cindy]

Hi, David,

BTW, do you know how to find the minimum norm of the lattice? I posted a question regarding this in this group. Do you know which function I should use?

Thanks.

Cindy

On Wednesday, September 5, 2012 4:30:40 PM UTC+8, David Loeffler wrote: