Function: qfbclassno
Section: number_theoretical
C-Name: qfbclassno0
Prototype: GD0,L,
Help: qfbclassno(D,{flag=0}): class number of discriminant D using Shanks's
 method by default. If (optional) flag is set to 1, use Euler products.
Doc: ordinary class number of the quadratic order of discriminant $D$, for
 ``small'' values of $D$.

 \item if  $D > 0$ or $\fl = 1$, use a $O(|D|^{1/2})$
 algorithm (compute $L(1,\chi_{D})$ with the approximate functional equation).
 This is slower than \tet{quadclassunit} as soon as $|D| \approx 10^{2}$ or
 so and is not meant to be used for large $D$.

 \item if $D < 0$ and $\fl = 0$ (or omitted), use a $O(|D|^{1/4})$
 algorithm (Shanks's baby-step/giant-step method). It should
 be faster than \tet{quadclassunit} for small values of $D$, say
 $|D| < 10^{18}$.

 \misctitle{Important warning} In the latter case, this function only
 implements part of \idx{Shanks}'s method (which allows to speed it up
 considerably). It gives unconditionnally correct results for
 $|D| < 2\cdot 10^{10}$, but may give incorrect results for larger values
 if the class
 group has many cyclic factors. We thus recommend to double-check results
 using the function \kbd{quadclassunit}, which is about 2 to 3 times slower in
 the range $|D| \in [10^{10}, 10^{18}]$, assuming GRH. We currently have no
 counter-examples but they should exist: we would appreciate a bug report if
 you find one.

 \misctitle{Warning} Contrary to what its name implies, this routine does not
 compute the number of classes of binary primitive forms of discriminant $D$,
 which is equal to the \emph{narrow} class number. The two notions are the same
 when $D < 0$ or the fundamental unit $\varepsilon$ has negative norm; when $D
 > 0$ and $N\varepsilon > 0$, the number of classes of forms is twice the
 ordinary class number. This is a problem which we cannot fix for backward
 compatibility reasons. Use the following routine if you are only interested
 in the number of classes of forms:
 \bprog
 ? QFBclassno(D) = qfbclassno(D) * if (D > 0 && quadunitnorm(D) > 0, 2, 1)
 ? QFBclassno(136)
 %1 = 4
 ? qfbclassno(136)
 %2 = 2
 ? quadunitnorm(136)
 %3 = 1
 ? bnfnarrow(bnfinit(x^2 - 136)).cyc
 %4 = [4]  \\ narrow class group is cyclic ~ Z/4Z
 @eprog\noindent Note that the use of \kbd{bnfnarrow} above is only valid
 because $136$ is a fundamental discriminant: that function is asymptotically
 faster (and returns the group structure, not only its order) but only supports
 \emph{maximal} orders.
 Here are a few more examples:
 \bprog
 ? qfbclassno(400000028) \\ D > 0: slow
 time = 3,140 ms.
 %1 = 1
 ? quadclassunit(400000028).no
 time = 20 ms. \\@com{ much faster, assume GRH}
 %2 = 1
 ? qfbclassno(-400000028) \\ D < 0: fast enough
 time = 0 ms.
 %3 = 7253
 ? quadclassunit(-400000028).no
 time = 0 ms.
 %4 = 7253
 @eprog\noindent See also \kbd{qfbhclassno}.
