The author has computed the class groups of all complex quadratic number fields
Q
(
−
D
)
Q(\sqrt { - D} )
of discriminant
−
D
- D
for
0
>
D
>
4000000
0 > D > 4000000
. In so doing, it was found that the first occurrences of rank three in the 3-Sylow subgroup are
D
=
3321607
=
prime
D = 3321607 = {\text {prime}}
, class group
C
(
3
)
×
C
(
3
)
×
C
(
9.7
)
(
C
(
n
)
C(3) \times C(3) \times C(9.7)\quad (C(n)
a cyclic group of order n), and
D
=
3640387
=
421.8647
D = 3640387 = 421.8647
, class group
C
(
3
)
×
C
(
3
)
×
C
(
9.2
)
C(3) \times C(3) \times C(9.2)
. The author has also found polynomials representing discriminants of 3-rank
⩾
2
\geqslant 2
, and has found 3-rank 3 for
D
=
6562327
=
367.17881
,
8124503
,
10676983
,
193816927
D = 6562327 = 367.17881,8124503,10676983,193816927
, all prime,
390240895
=
5.11.7095289
390240895 = 5.11.7095289
, and
503450951
=
prime
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
=
prime
D = 119191 = {\text {prime}}
, class group
C
(
15
)
×
C
(
15
)
C(15) \times C(15)
, had been found by A. O. L. Atkin; the next such D is
2075343
=
3.17.40693
2075343 = 3.17.40693
, class group
C
(
30
)
×
C
(
30
)
C(30) \times C(30)
. Finally,
D
=
3561799
=
prime
D = 3561799 = {\text {prime}}
has class group
C
(
21
)
×
C
(
63
)
C(21) \times C(63)
, the smallest D noncyclic for 3 and 7 together.