"Precision too low for generators, not given"

[Erick Knight]

Hello,

I am currently running some code computing the class groups of various fields.  This is going over a large number of fields and is occasionally returning the warning "precision too low for generators, not given."  If the only calls that I am invoking that are related to ideals are H.class_number() and H.ideal(3).prime_factors()[0].is_principal() (where H is the field under consideration), does this imply that any calculations are possibly incorrect or are being skipped, or are the class numbers and the determination of principality correct even knowing that the warning is happening?

Erick Knight

[Jeroen Demeyer]

I think it means that PARI didn't compute the unit group for certain
number fields. Since you don't need the unit group, I see no issue.

[John Cremona]

On 4 January 2017 at 20:31, Jeroen Demeyer <jdem...@cage.ugent.be> wrote:
> I think it means that PARI didn't compute the unit group for certain number
> fields. Since you don't need the unit group, I see no issue.

From what I know of the algorithm used -- and one should ask the pari
list to be certain -- it computed h*R (class number times regulator)
analytically and then, by factoring primes up to some bound, obtains
lower bounds on both h and R selarately until the product is within a
factor <2 of the known product. If that is the algirithm in use here,
then it could mean that when the unit generators are not certain then
neither is the class number.

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[Jeroen Demeyer]

Sorry, I was wrong. I actually looked at the PARI source code this time
and the warning comes from the bnfisprincipal() function to determine
the class of a given ideal in the class group (so, in particular, it can
be used to check whether an ideal is principal). This function can also
compute a generator of the ideal, but it might fail doing that, giving
the warning message. I believe this does not affect the correctness.

[John Cremona]

That sounds correct to me too, so thanks for checking the source code.
When it determines that an ideal is principal it can reconstruct the
generator using some additional information stored which includes logs
of elements in the various embeddings (as explained in Henri COhen's
second book), and at this point there may not be sufficient precision.
As Jeroen says, the correctness of the class number should be
affected.

[Erick Knight]

Thanks for both of your replies.  Just to double check so that I don't have any misunderstanding, PARI is correctly determining whether my ideal is principal, and if it is, it is attempting to produce a generator.  It may fail at that, but, since I never ask it for a generator, this produces no incorrect outputs.  Is that where this warning comes from and what I should make of it?

EK 

[Jeroen Demeyer]

On 2017-01-05 13:17, Erick Knight wrote:
> Thanks for both of your replies. Just to double check so that I don't
> have any misunderstanding, PARI is correctly determining whether my
> ideal is principal, and if it is, it is attempting to produce a
> generator. It may fail at that, but, since I never ask it for a
> generator, this produces no incorrect outputs. Is that where this
> warning comes from and what I should make of it?

Yes, I think that's right.