Numerical Results on Class Groups of Imaginary Quadratic Fields

Using techniques described in [3], we have computed the class number and class group structure of all imaginary quadratic fields with discriminant Δ for 0 < |Δ| < 1011. A novel verification algorithm based on the Eichler Selberg Trace Formula [15] was used to ensure that the correctness of our results does not rely on any unproved hypothesis. We present the results of our computations, and remark on specific evidence that was found pertaining to a number of heuristics. In particular, we present data which supports some of the Cohen-Lenstra heuristics [8], Littlewood’s bounds on L(1,χ) [14], and Bach’s bound on the maximum norm of the prime ideals required to generate the class group [1].

[1]  Michael J. Jacobson,et al.  Experimental Results on Class Groups of Real Quadratic Fields , 1998, ANTS.

[2]  Karim Belabas,et al.  On quadratic fields with large 3-rank , 2004, Math. Comput..

[3]  J. E. Littlewood,et al.  On the Class-Number of the Corpus P(√−k) , 1928 .

[4]  Johannes A. Buchmann,et al.  On some computational problems in finite abelian groups , 1997, Math. Comput..

[5]  D. Buell.  The last exhaustive computation of class groups of complex quadratic number fields , 1999 .

[6]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[7]  E. Bach.  Explicit bounds for primality testing and related problems , 1990 .

[8]  Johannes A. Buchmann,et al.  Computing the structure of a finite abelian group , 2005, Math. Comput..

[9]  Jeffrey Shallit,et al.  Algorithmic Number Theory , 1996, Lecture Notes in Computer Science.

[10]  Duncan A. Buell.  Small class numbers and extreme values of $L$-functions of quadratic fields , 1977 .

[11]  René Schoof,et al.  Hecke operators and the weight distributions of certain codes , 1991, Journal of Combinatorial Theory.

[12]  Franck Leprévost.  Courbes modulaires et 11-rang de corps quadratiques , 1993, Exp. Math..

[13]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[14]  Henri Cohen,et al.  Heuristics on class groups of number fields , 1984 .

[15]  Michael J. Jacobson.  Applying sieving to the computation of quadratic class groups , 1999, Math. Comput..