Class Groups of Quadratic Fields

Abstract

The author has computed the class groups of all complex quadratic number fields $Q(\sqrt { - D} )$ of discriminant $- D$ for $0 < D < 4000000$. In so doing, it was found that the first occurrences of rank three in the 3-Sylow subgroup are $D = 3321607 = {\text {prime}}$, class group $C(3) \times C(3) \times C(9.7)\quad (C(n)$ a cyclic group of order n), and $D = 3640387 = 421.8647$, class group $C(3) \times C(3) \times C(9.2)$. The author has also found polynomials representing discriminants of 3-rank $\geqslant 2$, and has found 3-rank 3 for $D = 6562327 = 367.17881,8124503,10676983,193816927$, all prime, $390240895 = 5.11.7095289$, and $503450951 = {\text {prime}}$. The first five of these were discovered by Diaz y Diaz, using a different method. The author believes, however, that his computation independently establishes the fact that 3321607 and 3640387 are the smallest D with 3-rank 3. The smallest examples of noncyclic 13-, 17-, and 19-Sylow subgroups have been found, and of groups noncyclic in two odd p-Sylow subgroups. $D = 119191 = {\text {prime}}$, class group $C(15) \times C(15)$, had been found by A. O. L. Atkin; the next such D is $2075343 = 3.17.40693$, class group $C(30) \times C(30)$. Finally, $D = 3561799 = {\text {prime}}$ has class group $C(21) \times C(63)$, the smallest D noncyclic for 3 and 7 together.