Query LMFDB to verify number theory results, including number field properties, elliptic curve data, and algebraic number theory computations.
Use lmfdb CLI to look up concrete examples and verify results against the LMFDB database.
brew tap frankieew/tap && brew install lmfdb-cli
# or build from source: go build -o lmfdb ./cmd/lmfdb
nf)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 |
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
Format: d.r.D.n where:
d = degreer = number of real embeddings (0 or positive)D = absolute discriminantn = 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)
-fKey fields in nf_fields collection:
label — LMFDB labeldegree — [K:Q]disc — discriminant Δ_Kclass_number — h(K)class_group — structure of Cl(K)signature — [r₁, r₂]regulator — regulator R_Kgalt — Galois group (transitive group label)subfields — subfield labelszk — integral basiscoeffs — minimal polynomial coefficientsec)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 |
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
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.
--browser flag. Run lmfdb install-browser first to download Chromium.--fmt json when you need to process results programmatically.-n and --offset to page through large result sets.-f to request only the fields you need — makes output cleaner and queries faster.