Difficulty with binary quadratic equation solver

[Kurt Foster <drsardonicus@earthlink.net>]

We were recently given the challenge of writing a GP script to solve  
the Diophantine binary quadratic equation

ax^2 + bxy + cy^2 + dx + ey + f = 0

where a, b, c, d, e, and f are integers.

Karim Belabas sketched out the procedure for the special cases, and  
Bill Allombert wrote a script for the "typical" case where b^2 - 4*a*c  
is not-a-square.

However, assuming gcd(a,b,c) = 1 and  b^2 - 4*a*c = m^2*N > 0 where m  
> 1 and N is a fundamental discriminant, there is an additional  
complication.  As in Bill's script, let

B = bnfinit(x^2 - b*x + a*c), and let U be the smallest power of B.fu  
with norm +1; that is,

U = B.fu if norm(B.fu) = 1, and U = (B.fu)^2 if norm(B.fu) = -1.

The difficulty is that in this case the binary quadratic form

x^2 + bxy + a*cy^2

describes norms of elements of the quadratic suborder O of index m in  
B.zk, and U might not be in O.  Now Bill's script checks certain  
congruence conditions for

(+/-)L[i]

where the L[i] are elements of B.zk of desired norm, produced by  
bnfisntnorm().  But if U^k is the smallest positive power of U which  
lies in O, one has to check

(+/-)U^j*L[i], for 0 <= j <= k-1.

which the script doesn't do if k > 1.

Bill provides the example

[a,b,c,d,e,f] = [1,0,-5,1,1,2]

for which the field discriminant N = 5, the index m is 2, U =  
(B.fu)^2, the smallest value of k is 3, and plowing through the  
calculations by hand shows that there are two "inequivalent" solutions

[x,y] = [1,1] and [-5, -2]

For a given positive fundamental discriminant N and index m > 1,  
determining the exact smallest value of k for which U^k is in O has  
proven to be rather exasperating, although certain multiples of it are  
easy to find.  I've dug into it a fair bit, but I'm sure this is a  
known calculation.  Can someone provide a reference?

Once the correct value of k is in hand, one can turn to the problem of  
doing the additional checking indicated above.  Unfortunately, my  
programming ability is practically nil...