Degree 4 number field counting discrepancy

[Qiyao Yu]

Dear LMFDB,

My coauthor (cc’ed) and I are trying to use LMFDB number field database to get some data on the number of degree 4 fields (with Galois group D4) ordered by discriminants bounded by |x|. We have noticed some discrepancies between the number provided by the database and those from a paper by Cohen-Diaz Y Diaz-Oliver.

The paper we are referring to is Counting discriminants of number fields (Journal de Théorie des Numbers, 2006). In table 7.2 on p587, the column N_{0,2}(D4, X) denotes the number of totally imaginary number fields of degree 4 (Galois group D4) with discriminants bounded by |x|. 

Starting from X=10^8, the number provided in Cohen et. al. is significantly larger than the result given by the LMFDB search.

For example, for 10^8, Cohen gives 3734826 and LMFDB gives 1313459. The difference persists as X gets larger.

I understand that the two sources agree below 10^7 because there exists a complete list of fields with discriminant bounded below. For bigger discriminants, is there any way to address this discrepancy?

Thank you!

Best,

Qiyao Yu

[John Jones]

The LMFDB has a page describing under which criteria, the number field data is claimed to be complete:

  https://www.lmfdb.org/NumberField/Completeness

It does not claim to have all D_4 fields up to 10^8.

John Jones

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[Qiyao Yu]

Hi John,

Thank you so much for your prompt reply! In case I have further questions I will reach out again.

Best,

Qiyao