New Data Submission: Degree 5, 7, 11, and 13 Totally Real Number Fields with Single Prime Ramification
Shen, Isaac
Dear LMFDB Editorial Team,
I am an undergraduate researcher currently investigating high-degree totally real number fields. While cross-referencing my results with the LMFDB, I discovered several gaps in degrees 5，7，11 and 13 that I have successfully filled using a custom search algorithm.
Key Findings:
l Degree 5: Generated over 140 Frobenius F5 fields; including 25 unindexed single-prime ramified extensions within the 10000^3 < ∆ < 25000^3 range, successfully filling blind spots in the current LMFDB.
l Degree 7: Isolated about 20 Frobenius F7 fields; including 5 unindexed single-prime ramified extensions and 15 targeted p^4 q^3 discriminant structures via a novel grafting method.
l Degree 11: Discovered 14 unique fields under extreme constraints.
l Degree 13: Trapped 5 exotic non-abelian fields with exact single-prime ramification, tracking highly elusive structures across C13⋊C3 (∆ =1063^8, 1459^8), C13⋊C4 (∆=2617^9), C13⋊C6 (∆=3469^10), and the massive F13 Frobenius extension (∆=4729^11).
All findings have been verified using PARI/GP (and cross-checked with Magma for n>=13).
Methodology:
The underlying mathematical engine bypasses the immense computational complexity of direct high-degree searches by utilizing a sophisticated top-down class field tower reduction:
- Base Field Scans: First, the pipeline searches for lower-degree base fields (e.g., degree 5 or 6) whose class group possesses a structure isomorphic to
or 
. - Unit Root Embedding: Embed the 11th or 13th roots of unity (
or 
) into the field extension to filter out the target subfields with the precise Galois behavior. - Algebraic Valuation & Power Criteria: Locate a defining polynomial whose constant term is a perfect 11th or 13th power. The product of roots reducing to the subfield must evaluate to a perfect 11th or 13th power in that local ring, while ensuring the individual roots themselves are not trivial powers.
- Radical Extension & Absolute Reduction: Take the 11th or 13th radical of the polynomial's root, isolate the target 11 or 13-degree subfield, and compute the final optimized defining polynomial using the polredabs() algorithm in PARI/GP.
Would you be interested in incorporating these results into the database? I am happy to provide the full data and verification scripts upon your request.
Thank you very much for your kind attention. I look forward to contributing to this global number theory community.
Best regards,
Isaac (Enyang) Shen
Junior Computer Science Student
Cedarville University
Ohio USA
John Jones
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