Querying LMFDB for Mathematical Verification

Use lmfdb CLI to look up concrete examples and verify results against the LMFDB database.

Installation

brew tap frankieew/tap && brew install lmfdb-cli
# or build from source: go build -o lmfdb ./cmd/lmfdb

Number Fields (nf)

Quick Reference

lmfdb nf [options]

Flag	Meaning	Example
-d <n>	Degree [Q(α):Q]	-d 3 (cubic fields)
-n <n>	Number of results	-n 50
--disc <val>	Discriminant	--disc -23
--class <n>	Class number h(K)	--class 1
--sig <r1,r2>	Signature (r₁,r₂)	--sig 0,1 (totally imaginary)
--id <label>	Specific field by LMFDB label	--id 2.0.3.1
--sort <field>	Sort (prefix - for desc)	--sort -disc
-f <fields>	Select specific fields	-f label,disc,class_number
--fmt json	Output as JSON	--fmt json
-o <file>	Save to file	-o results.csv --fmt csv
--browser	Bypass reCAPTCHA	for large queries

Common Verification Tasks

Find all quadratic fields with class number 1:

lmfdb nf -d 2 --class 1 -n 100

Check imaginary quadratic fields (signature (0,1)):

lmfdb nf -d 2 --sig 0,1 -n 50

Look up a specific number field by label:

lmfdb nf --id 2.0.3.1
# Returns full record: discriminant, class number, Galois group, regulator, etc.

Cubic fields sorted by discriminant:

lmfdb nf -d 3 --sort disc -n 20

Totally real quartic fields:

lmfdb nf -d 4 --sig 4,0 -n 20

Export for further analysis:

lmfdb nf -d 2 -n 200 -f label,disc,class_number,regulator --fmt json -o quadratic_fields.json

LMFDB Number Field Labels

Format: d.r.D.n where:

• d = degree
• r = number of real embeddings (0 or positive)
• D = absolute discriminant
• n = index among fields with same (d, r, D)

Example: 2.0.3.1 = degree 2, no real embeddings, |disc| = 3, first such field = Q(√-3)

Available Fields for -f

Key fields in nf_fields collection:

• label — LMFDB label
• degree — [K:Q]
• disc — discriminant Δ_K
• class_number — h(K)
• class_group — structure of Cl(K)
• signature — [r₁, r₂]
• regulator — regulator R_K
• galt — Galois group (transitive group label)
• subfields — subfield labels
• zk — integral basis
• coeffs — minimal polynomial coefficients

Elliptic Curves (ec)

Quick Reference

lmfdb ec [options]

Flag	Meaning	Example
-r <n>	Mordell-Weil rank	-r 2
-t <n>	Torsion order	-t 7
--conductor <n>	Conductor N	--conductor 11
-n <n>	Number of results	-n 50
--sort <field>	Sort	--sort conductor
-f <fields>	Select fields	-f lmfdb_label,rank,conductor

Common Verification Tasks

Find rank 2 elliptic curves:

lmfdb ec -r 2 -n 20

Curves with torsion group of order 7:

lmfdb ec -t 7 -n 10

Curves of conductor 11:

lmfdb ec --conductor 11 --fmt json

Export rank data for analysis:

lmfdb ec -r 0 -n 100 -f lmfdb_label,conductor,torsion_structure --fmt csv -o rank0_curves.csv

Other Collections

lmfdb list   # Show all available collections

Available: number fields, elliptic curves, genus 2 curves, Dirichlet characters, Maass forms, modular forms, local fields, Artin representations, Belyi maps.

Tips

• reCAPTCHA blocking: For large queries (n > 50), use --browser flag. Run lmfdb install-browser first to download Chromium.
• JSON for parsing: Use --fmt json when you need to process results programmatically.
• Pagination: Combine -n and --offset to page through large result sets.
• Specific fields: Use -f to request only the fields you need — makes output cleaner and queries faster.