There are two reasons why people work with non-maximal orders: because they're actually interested in their arithmetic; or (more often) because they're working with examples where the discriminant is too large to efficiently factor. Which is the case in your problem? In the example you give, you're letting PARI choose the order for you, but the discriminant is small enough that one can find a maximal order in a few milliseconds anyway (and Sage is perfectly happy to work with residue fields of maximal orders).
If you genuinely don't want to factor the discriminant, but the orders you're interested in are "monogenic" (generated by a single element over Z), then you can just factor the characteristic polynomial of the generator over Fp and that gives you the factorisation, and the reduction maps, immediately. Can you give an example of a case you'd be interested in where the maximal order is really too big to easily find? Regards, David On 11 March 2016 at 14:42, Misja <[email protected]> wrote: > For a number field N I am trying to factor an integral prime p in a > p-maximal order Op. In the end I would like a map from the quotient of the > p-maximal order Op/P (for P|p) to some finite field in Sage's standard > finite field form, but I can't quite figure out how to do it. > > Firstly, Sage doesn't have ideals or residue fields of non-maximal orders > implemented, so I am trying to compute the factorisation of p in Op via the > PARI/GP interface with sage. I think this can be done, for example, in the > following way, in which we compute a 7-maximal order of the number field > defined by x^8+x^3-13*x+26 and compute all primes above 7 in this order. > > x=polygen(QQ); > > nf=gp.nfinit([x^8 + x^3 - 13*x + 26,[7]]); #Note the [7] at the end, > indicating a 7-maximal order only. > nf; > > fact=gp.idealprimedec(nf,7); > fact; > > res_field_sizes=[]; > for i in fact: > res_field_sizes.append(i[1]**i[4]); > > res_field_sizes; > > Now, for each element fact[i] of fact, I would like to compute a > homomorphism Op->Op/fact[i]->GF(res_field_sizes[i-1],'gen') - note that > we're using a vector in pari/gp and a list in sage hence fact[i] > corresponds to res_field_sizes[i-1]. I can't quite get this to work. > > I feel like there are two options: > > 1. Translate the p-maximal order and prime factorisation back to sage. > Take a poly quotient or something. Map this to a finite field. > 2. Keep on working in pari/gp and use, for example, nfeltreducemodpr > to reduce elts of Op to the residue field (does this really calculate Op/P > if the nf.zk is a non-maximal Z-basis?). Map this to a finite field > somehow. Translate back to Sage. > > I can't really get either approach to work at the moment. But I wouldn't > call myself a big Sage or PARI/GP expert, so it is very possible that I am > missing something. > > Any suggestions would be much appreciated! > > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
