2011/10/09 by Jonathan Hanke, Hanke, Jonathan
Computer Science · Mathematics · #11E08 #11E12 #11E41 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT #msc:11E08 #msc:11E12 #msc:11E41
paper · pdf · doi:10.48550/arxiv.1110.1876
20 pages, 12 tables
arxiv created 2011/10/09 · openalex publication_date 2011/10/09 · arxiv updated 2011/10/11 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
In this paper we give an algorithm for enumerating all primitive (positive) definite maximal Z-valued quadratic forms Q in n >= 3 variables with bounded class number h(Q) <= B. We do this by analyzing the exact mass formula [GHY], and bounding all relevant local invariants to give only finitely many possibilities. We also briefly describe an open-source implementation of this algorithm we have written in Python/Sage which explicitly enumerates all such quadratic forms of bounded class number in n >= 3 variables. Using this we determine that there are exactly 115 primitive positive definite maximal Z-valued quadratic forms in n >= 3 variables of class number one, and produce a list of them. In a future paper we will complete this chain of ideas by extending these algorithms to allow the enumeration of all primitive maximal totally definite OF-valued quadratic lattices of rank n >= 3, where OF is the ring of integers of any totally real number field F.