Method is durable; catalogue data has a shelf life
KEVOS separates two kinds of knowledge. Method — definitions, algorithms, complexity results, decision criteria — is durable and belongs in the library. Catalogue data — tables of class numbers, fundamental units, curve coefficients — is a snapshot: it can be extended, corrected and superseded, and transcription introduces silent corruption. This library therefore documents how each table is computed and interpreted, and points to a live authoritative source for the values themselves.
Learning objectives
- State the two-layer architecture and the reasoning behind it.
- Identify the risks specific to transcribing numeric tables.
- Locate the current authoritative source for each class of data.
- Verify a value obtained from an external database.
Section 01The two-layer architecture
Definitions, algorithms with their preconditions, complexity results, failure modes, verification procedures and selection criteria. These do not expire: the Round 2 algorithm works the same today as when it was published.
Class number tables, unit and regulator tables, curve tables, discriminant lists. These are extended and corrected continually, and any static copy begins going out of date immediately.
If a reader needs to know how a quantity is computed and what it means, the answer is here. If they need the value for a specific discriminant, the answer is a query against a live database — and this library says which one and how to interpret what comes back.
Section 02Why transcription is the wrong choice
Silent corruption
Digit substitution, dropped decimal points and collapsed columns produce values that are plausible and wrong. A corrupted class number carries no diagnostic signal — it is simply an integer of about the right size.
Data currency
Tables computed decades ago predate later corrections, extended ranges and improved certification. A value that was conditional then may be unconditional now, or may have been revised.
Loss of provenance
A transcribed number loses its conditionality, its precision and the algorithm that produced it. A regulator without its certified precision cannot be compared with anything.
Coverage
Any printed table stops somewhere. A live database has been extended enormously beyond the ranges that could be printed, and continues to grow.
Licensing
Substantial tables from published works are protected. Republishing them is neither necessary nor permissible.
Linking instead
A pointer to a live source is always current, carries full provenance and metadata, and usually offers far more than the original table did.
An error in a numeric table does not announce itself. It propagates into every downstream calculation, each of which succeeds and produces a well-formed result. This is the same failure mode that governs the rest of the subject, and it is the reason KEVOS treats catalogue data as a sourcing question rather than a content question.
Section 03Where to source each class of data
| Data | Source | Notes |
|---|---|---|
| Class numbers and class groups of quadratic fields | LMFDB; PARI/GP quadclassunit | Computed on demand for arbitrary discriminants, with conditionality reported |
| Number field tables by degree and signature | LMFDB number field database | Includes defining polynomial, discriminant, class group, unit rank and Galois group |
| Fundamental units and regulators | PARI/GP bnfinit; Magma | Report the precision and the proof flag with the value |
| Elliptic curves over ℚ | LMFDB; the Cremona database | Indexed by conductor; includes rank, torsion, Tamagawa numbers and generators |
| Modular forms and L-functions | LMFDB | Cross-referenced to the associated curves and fields |
| Factorisations of numbers of special form | The Cunningham Project tables | Maintained continuously; the reference for re ± s |
| Prime tables and records | The PrimePages and associated record lists | Includes certificates for record primes |
For a single discriminant it is usually faster and more reliable to compute the invariant directly in PARI/GP than to search a table — and the computation reports its own conditionality, which a table lookup may not.
Section 04Verifying an external value
- Stage 01Record the queryNote the database, the version or access date, and the exact identifier used.
- Stage 02Check the conventionsConfirm the discriminant convention, the equivalence convention for forms, and whether the regulator includes any index factor.
- Stage 03Recompute independentlyRecompute the value in a system with an independent code path, and compare.
- Stage 04Apply a structural checkVerify the fundamental identity for prime decomposition, the analytic class number formula for h and R, or Mazur's theorem for a torsion subgroup.
Two databases may share an upstream source, so agreement between them is weaker evidence than it appears. A check against an independent mathematical identity — hR against the analytic formula, or ∑eifi = n — tests the value itself rather than its provenance.
ReferenceFrequently asked questions
Does this policy apply to worked examples?
No. Small illustrative examples computed and verified within a page are method, not catalogue data, and they belong here. The policy concerns systematic tables of values that a reader would otherwise treat as a reference dataset.
What if an external source disagrees with a computation?
Check conventions first, since most disagreements are convention mismatches rather than errors. If a genuine discrepancy survives, apply a structural check to determine which value is wrong, and report it to the database maintainers — corrections are how these resources stay reliable.
Why not cache a snapshot for offline use?
A cache is acceptable provided it records its retrieval date and is treated as a snapshot rather than as authority. The failure this policy guards against is a table that has lost its provenance and is subsequently trusted as though it were current.
NavigateContinue in this stream
Curated next steps from this page. The site also surfaces algorithmically related reading below.
ProvenanceSources and further reading
This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.