Independent research · Austin

A table of 124,693 degree-24 fields, built on 18 August 2026. Each record separately reports the group-label evidence, polynomial reduction, discriminant calculation and maximal-order status.

For the complete downloads and board data, see the data page. Use the lookup below to inspect an individual cell.

124,693 degree-24 fields

Table built 18 August 2026, 10:08 UTC

124,693 rows over 20,882 of the 25,000 transitive groups of degree 24, covering 78,703 distinct cells, where a cell is a group together with a number of real roots. Every row is an irreducible degree-24 polynomial, and 124,242 of them are in canonical polredabs form, so they match against any other table without further work. Each row states on four separate axes how far it has been certified rather than asserting a single verdict.

The published table is 124,693 rows over 20,882 groups, covering 78,703 cells. On the group-name axis all 124,693 rows are evidence at a stated margin and none is exact. On canonical form, 124,242 rows are polredabs and 451 are not. On the discriminant axis, 31,664 are unconditional, 92,672 are conditional on a cofactor being squarefree, and 357 are undetermined.

Drawn to scale, so the two slivers are as small as they are. A single verdict per row would have to report the weakest axis and would hide the other three.

What the margin on the group axis measures: a row asserts that its group's class-rate law fits the observed Frobenius cycle types better than each of the other 24,999 groups, by that many nats. The calibration of the reader is published beside the table.

We recomputed group labels from the polynomial coefficients before publication. Where the bank’s existing label disagreed with that reread, the published table uses the reread. The margin records the statistical evidence for the label; it is not an exact Galois-group certificate.

Download

fields.jsonl.gz is the whole table. The browsable copy is a hundred groups per file, one best-certified witness per cell, indexed by by-group/index.json. summary.json holds the counts, and MANIFEST.json the hash of each file. Those hashes are reproducible rather than merely published: the table and every shard are byte-identical across builds from the same source. The row format and the checker are at what to run.

shasum -a 256 fields.jsonl.gz    # compare against MANIFEST.json

Look up a field

Enter a group number and real-root count to retrieve a polynomial and the checks recorded for its field. The lookup loads the relevant hundred-group shard and returns the table’s best-certified witness for that cell.

24T signature

Group names

Open-cell counts come from the dated organizer board; this file records published fields. 78,703 cells here carry a published field. A cell name is written n_r, so 1_0 is 24T1 with no real roots and 1_24 is the same group totally real.

SHA-256 of cells.txt

1e341f2036327df442c77ce1dbafd17b08d307acd46ef989a0fc0c26658f2e0f

Pulled 16 August 2026, 11:54 UTC, hashed at 12:20. Independent timestamp 16 August 2026, 12:19:30 UTC, RFC 3161 from FreeTSA, and verification on that token returned OK.

cells.txt · HASH.txt · timestamp token · FreeTSA CA · FreeTSA TSA

shasum -a 256 cells.txt
openssl ts -verify -data cells.txt \
  -in OUR_CELLS.freetsa.tsr \
  -CAfile freetsa_cacert.pem \
  -untrusted freetsa_tsa.crt

What this is not

No group name anywhere here is a certified Galois group computation. A name on the list means a field existed for that cell at the pull.