A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

72CHAPTER 10. DISCRIMANNTS, NORMS, AND FINITENESS OF THE CLASS GROUP
if h eq 1 then
printf "%o, ", d;
end if;
end if;
end for;
1, 2, 3, 7, 11, 19, 43, 67, 163
It is a theorem that the above list of 9 fields is the complete list with class number 1.
More generally, it is possible (in theory), using deep work of Gross, Zagier, and
Goldfeld involving zeta functions and elliptic curves, to enumerate all quadratic
number fields with a given class number.