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.
