Gerhard Niklasch on Fri, 19 Jun 1998 16:19:07 +0200 |
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Re: Class groups |
In response to > Message-Id: <9806191249.AA00260@rhrz.uni-bonn.de> > From: Michael Stoll <Michael_Stoll@math.uni-bonn.de> > Date: Fri, 19 Jun 1998 14:49:12 +0200 > > Gerhard Niklasch <nikl@mathematik.tu-muenchen.de> writes > > GN> Anybody know which version of the PARI kernel would be included in that > GN> version of MAGMA? > The number field algorithms of Magma are based on KANT, not on PARI. Thanks to John Cannon for clearing this up in an earlier email; I guess I had been reading your (Michael's) first posting too fast and believed I saw some PARI function names appearing in what I took to be MAGMA diagnostic output. Sorry for any confusion I may have caused. [...] > In what way could the result be wrong? Or does `not guaranteed' mean that > there might be no result at all? E.g. if we don't search sufficiently far (where `sufficiently' depends on whether one believes in GRH or not ;^), we might have found generators only for a proper subgroup of the class group, so (assuming we caught all relations) our computed h will be a proper divisor of the true value, and we might also have found only a proper subgroup of the group of units, and thus the computed regulator R will be an integral multiple of the true value. (If we might have missed relations, it'll be even worse!) Things may conspire to give the correct product hR, and thus fail to indicate that anything might be wrong. If some veteran user of one of the systems under discussion has seen and archived (or could reproduce) an example where this actually happens, perhaps with contrived parameters, I think it would be valuable if they could post it say here on pari-dev or on pari-users to spread the knowledge, and to warn later generations not to trust their machines too blindly. As far as I could tell (without correlating the franglais messages to the code issuing them) from what gp was doing with x^4+5*239*x^2+5*239^2 at default parameter settings, the problem there was that it kept missing a relation, and thus the computed class group was always a double cover of the true one, and hR was twice as large as it should have been. So it noticed that something was wrong, and kept increasing the size of the factor base. (I may be wrong -- corrections welcome.) [...] > My machine has a Pentium 200 S CPU, in case you want to compare timings > (timings vary, however, because of some probabilistic parts of the > algorithms, I guess). This is easily taken care of by initializing the random seed explicitly (setrand() in gp) before embarking on the computation. Thanks for the update, Gerhard