Improved methods for finding imaginary quadratic fields with high 𝑛-rank
Bibliographic record
Abstract
We describe a generalization and improvement of Diaz y Diaz’s search technique for imaginary quadratic fields with 3 3 -rank at least 2, one of the most successful algorithms for generating many examples with relatively small discriminants, to find quadratic fields with large n n -ranks for odd n ≥ 3 n \geq 3 . An extensive search using our new algorithm in conjunction with a variety of further practical improvements produced billions of fields with non-trivial p p -rank for the primes p = 3 , 5 , 7 , 11 p = 3, 5, 7, 11 and 13 13 , and a large volume of fields with high p p -ranks and unusual class group structures. Our numerical results include a field with 5 5 -rank at least 4 with the smallest absolute discriminant discovered to date and the first known examples of imaginary quadratic fields with 7 7 -rank at least 4.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.017 | 0.009 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".