William Hart on Fri, 04 Jan 2008 17:40:42 +0100 |
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Relative number fields question |
Hi all, Given a degree three abelian extension L/K of a quadratic field K, I am trying to find an ideal QL in L above a given prime ideal Q in K. I set up my base field K, and the relative extension L/K, find an absolute equation for L/Q and define the number field "Labs" to be the corresponding absolute number field: K=nfinit(y^2 + 939636383790*y + 226927609420691962278648); g=x^3 + Mod(11067, y^2 + 939636383790*y + 226927609420691962278648)*x^2 + Mod(23329236, y^2 + 939636383790*y + 226927609420691962278648)*x + Mod(-1/3*y - 193638491109, y^2 + 939636383790*y + 226927609420691962278648); L=rnfinit(K,g); Leqn=rnfequation(K,g,1)[1]; Labs=nfinit(Leqn); I find the prime ideal factors of a certain principal ideal in K and choose one of them, calling it Q. This is a prime ideal of K: Afac=idealfactor(K,[2,-2]~); Q=Afac[1,1]; Now I want an ideal in L that is above Q, i.e. I want to factor Q in L. There are a number of strategies. One is to find the norm Qn of Q, which should be a power of a prime p, factor p into prime ideals Qn_i in Labs then drop each down into K until I find the one that is above Q. Qn=idealnorm(KQ); Qn=factor(Qn)[1,1]; Qnfac=idealprimedec(Labs,Qn); fac=idealhnf(Labs,Qnfac[1]); Qndown=rnfidealdown(L,fac); But Pari responds: *** rnfidealdown: incorrect type in rnfidealabstorel. My original code attempted to go up from the ideal Q into L and then factor that: QL=rnfidealup(L,Q); But if I now do: nfisideal(Labs,QL) it returns false. The following code does something (though doubtlessly not the right thing): QL=mathnf(Mat(nfalgtobasis(Labs,QL))); Ql=idealfactor(Labs,QL)[1,1]; Qmodpr=nfmodprinit(Labs,Ql); nfeltreducemodpr(Labs,Mod(1,Labs.pol),Qmodpr); But it responds: *** significant pointers lost in gerepile! (please report). Can anybody tell me what I am doing wrong!? Is this last error a bug, and should I report it? Is there an easy way to find an ideal of Labs above my ideal Q? I'm using the latest pari/gp development snapshot due to a known bug elsewhere in the stable release which kills other code I am using. Regards, Bill Hart University of Warwick ____________________________________________________________________________________ Never miss a thing. Make Yahoo your home page. http://www.yahoo.com/r/hs