PS: In your example, the order generated by the element b = 3033120361/143075599831793664*a^4 + 1953773/876688724459520*a^3 + 53/38811740405760*a^2 + 1/3158507520*a
is actually locally maximal at 7, so you can compute all the primes above 7 and the residue maps for this order just by computing the min poly of b and factorising it mod 7. Amusingly, the factorisation of the minimal polynomial turns out to be sage: b.minpoly().change_ring(GF(7)).factor() (x + 2) * (x + 3) * (x + 4) * (x + 5) * (x + 6) David On 26 March 2016 at 14:53, David Loeffler <d.a.loeff...@warwick.ac.uk> wrote: > Dear Misja, > > What I had in mind was something like this. Given some monstrous number > field K with enormous discriminant, and some small prime p, you can ask > Sage for a p-maximal order and it'll find one reasonably quickly, as you > know. > > All I was saying is that if the resulting order O is of the form Z[X] / > (f), for some polynomial f in Z[X] -- i.e. if "O.ring_generators()" is a > list of length 1 -- then you can quickly compute all possible maps O --> > Fp-bar by just factoring f modulo p. (Sage really ought to have this coded > up already; sadly it doesn't, but it's trivial to implement.) Maybe this > isn't a terribly helpful remark; I don't know how likely it is that the > orders you work with have this form. > > David > > On 17 March 2016 at 16:33, Misja <m.steinm...@gmail.com> wrote: > >> Hi David, >> >> Thank you very much for this helpful reply! You're right, of course: my >> example was silly. Here's an example with a 104 digit discriminant that my >> code just got stuck on (for a bit). >> >> x=polygen(QQ); >> K=NumberField(x^5 + 16255223088*(x^4) - 330681713908949415936*(x^3) - >> 5058938091171222191449571328000*(x^2) - >> 2907488578277274989701398473183198183424*x + >> 76587534178613500902724649685949865805676749520896,'a'); >> fact=K.factor(5); >> print K.residue_field(fact[0][0]); >> >> What I am trying to do is to calculate reductions of modular forms modulo >> some prime ideal P. However, the coefficients of such a modular form may >> lie in a number field with a 1000 digit discriminant and I want to avoid >> trying to factor this if I can, hence my trying to use p-maximal orders. >> >> Could you explain a bit more what you mean in the monogenic case? Could >> you do something like EquationOrder(K.defining_polynomial(),'alpha'), take >> a p-maximal order there and then do what you are suggesting? Although, >> actually, I don't know if sage can calculate a p-maximal order of a given >> order. >> >> Misja >> >> >> >> On Thursday, 17 March 2016 15:50:59 UTC, David Loeffler wrote: >>> >>> 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 <m.ste...@gmail.com> 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 sage-support...@googlegroups.com. >>>> To post to this group, send email to sage-s...@googlegroups.com. >>>> 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 sage-support+unsubscr...@googlegroups.com. >> To post to this group, send email to sage-support@googlegroups.com. >> 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. 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