Software, Tables and Sources
Published Tables of Fields and Curves
Standard tables of number fields, elliptic curves, class numbers and factorisations, and how to use them responsibly.
Engineering / MathematicsSoftware, Tables and Sources2 min readKV-MATH-0678
Extensive tables exist for number fields, elliptic curves, class numbers and factorisations. They save enormous effort and are the single best source of test material.
What is tabulated
| Table | Indexed by | Typical contents |
|---|---|---|
| Number fields | Degree, signature, discriminant | Defining polynomial, class group, regulator, Galois group |
| Elliptic curves | Conductor | Model, rank, torsion, reduction data |
| Class numbers | Discriminant | Class number and group structure for quadratic fields |
| Factorisations | Number of special form | Known factors of numbers such as those in the Cunningham project |
| Modular polynomials | Level | Coefficients for modular equations |
Using tables responsibly
Completeness claims
A table claiming to list all objects below a bound rests on a completeness argument, which is separate from the correctness of the individual entries.
Conditional entries
Building on tables
Using tabulated data
- Verify a sampleRecompute several entries independently before trusting a table.
- Record provenanceWhich table, which version, which conventions.
- Carry conditionalityA result depending on a conditional entry is itself conditional.
- Report discrepanciesA genuine disagreement is worth investigating and reporting.
Related resources
Software packages frequently ship with tabulated data built in — see software packages — which is usually the most convenient access route and carries the package's conventions.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — App. B. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.