Bill Allombert on Wed, 02 Sep 2009 16:06:49 +0200 |
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Re: A number fields question |
On Wed, Aug 19, 2009 at 10:49:47AM +0100, Slessor R. wrote: > > Dear all, > > I was wondering if you would be able to help me with a quick question? I have the following relative number fields extension L/K (also Galois): > > K=nfinit(y^2 + y + 1); > f= x^6 + Mod(3969*y+10584,y^2+y+1); > L=rnfinit(K,f); > lambda = Mod(Mod(-4/64827*y-1/43218,y^2+y+1)*x^5 + Mod(2/9261*y-1/9261,y^2+y+1)*x^4 + Mod(-1/294*y-1/441,y^2+y+1)*x^3 + Mod(1/441*y+1/147,y^2+y+1)*x^2 + Mod(-1/42*y-1/42,y^2+y+1)*x+1/3, x^6 + Mod(3969*y+10584,y^2+y+1)); > > > lambda has been chosen in such a way that it is totally real under the action of Gal(L/K) > > I would like to know if there is a way to get PARI to compute all elements x of the ring of integers O_L of L such that |Tr_{L/K}(lambda. \bar{x}. x)| = 1? > > where \bar{x} is just the usual complex conjugation. I would try the following: Compute an integral basis for O_L and consider q(x)=Tr_{L/K}(lambda. \bar{x}. x) as a quadratic form over O_L with value in C and compute its matrix. then you should be able to answer your question. Cheers, Bill.