Arno, Steven; Robinson, M. L.; Wheeler, Ferrell S. Imaginary quadratic fields with small odd class number. (English) Zbl 0904.11031 Acta Arith. 83, No. 4, 295-330 (1998). In this paper the authors determine all imaginary quadratic number fields of odd class number \(h\) in the range \(5\leq h\leq 23\). The results for \(h=1,2,3,4\) were already known. The authors employ techniques from Goldfeld, Oesterlé, Stark, Montgomery and Weinberger, and the first author. With those results it is straightforward to give upper bounds for the discriminant \(-d\) (\(d\equiv 3 \bmod 4\) a prime) of potential imaginary quadratic fields. The crucial part of the paper is the reduction of these huge bounds (\(10^{550}\) in the beginning) to a size of about \(10^{15}\). For this, the authors substantially refine an approach by S. Arno. They use a separation of the minima (minimal coefficients) of reduced quadratic forms of discriminant \(-d\) combined with earlier methods from Stark and Montgomery-Weinberger. The last step then consists of an enumeration procedure for determining all fields of given class number \(h\) below the reduced bound by sieving techniques. Reviewer: M.Pohst (Berlin) Cited in 1 ReviewCited in 14 Documents MSC: 11R29 Class numbers, class groups, discriminants 11R11 Quadratic extensions 11Y40 Algebraic number theory computations 11-04 Software, source code, etc. for problems pertaining to number theory 11H50 Minima of forms 11N36 Applications of sieve methods Keywords:imaginary quadratic fields with odd class numbers; separation of the minima; reduced quadratic forms; sieving techniques PDFBibTeX XMLCite \textit{S. Arno} et al., Acta Arith. 83, No. 4, 295--330 (1998; Zbl 0904.11031) Full Text: DOI EuDML Online Encyclopedia of Integer Sequences: Largest prime p==3 (mod 8) such that Q(sqrt(-p)) has class number 2n+1. Imaginary quadratic fields with class number 2 (a finite sequence). Discriminants of imaginary quadratic fields with class number 3 (negated). Discriminants of imaginary quadratic fields with class number 5 (negated). Discriminants of imaginary quadratic fields with class number 6 (negated). Discriminants of imaginary quadratic fields with class number 7 (negated). Discriminants of imaginary quadratic fields with class number 9 (negated). Discriminants of imaginary quadratic fields with class number 11 (negated). Discriminants of imaginary quadratic fields with class number 13 (negated). Discriminants of imaginary quadratic fields with class number 15 (negated). Discriminants of imaginary quadratic fields with class number 17 (negated). Discriminants of imaginary quadratic fields with class number 19 (negated). Discriminants of imaginary quadratic fields with class number 21 (negated). Discriminants of imaginary quadratic fields with class number 23 (negated). Numbers n such that Q(sqrt(-n)) has class number 4.