Computing the class number of $\mathbb{Q}(\sqrt{1533157})$ I am trying to compute the class number of $\mathbb{Q}(\sqrt{1533157})$ in Magma. Can anyone explain why it's taking so long to compute? I'm currently running Magma V2.18-7. Below is my code:
SetClassGroupBounds("GRH"); 
K := QuadraticField(1533157);
ClassNumber(K);

 A: EEDDIITTT: I KNOW the final answer is four!!!!!! My viewpoint gives eight and then identifies pairs, resulting in four. See quote from blog at the end!!!
When this first came up I had not written the class number program yet: the class group of indefinite binary quadratic forms has eight elements, for the number field you just identify $(a,b,c)$ with $(-a,b,-c)$ and get down to four elements. I discuss the explicit mapping from forms (triples) to ideal thingies in a few places in http://math.blogoverflow.com/2014/08/23/binary-quadratic-forms-over-the-rational-integers-and-class-numbers-of-quadratic-%EF%AC%81elds/
 1533157    factored   23 * 191 *  349

    1.             1        1237        -747   cycle length            68
    2.            -1        1237         747   cycle length            68
    3.             3        1237        -249   cycle length            70
    4.            -3        1237         249   cycle length            70
    5.            71        1221        -149   cycle length            78
    6.           -71        1221         149   cycle length            78
    7.           149        1221         -71   cycle length            78
    8.          -149        1221          71   cycle length            78

  form class number is   8

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$

EEEEEDDDIIIIITTTTTTT: Quoting from my BBBBBBLLLLLLOOOOOOOGGGG:
So, the mapping to ideals says  IDENTIFY $ \langle A,B,-C \rangle  $ and
$ \langle -A,B,C \rangle  ,$ send them to the same ideal. Why not? We already have an equality of ideals in 
$$ \left[ A, \frac{B + \sqrt \Delta}{2} \right]  =  \left[ -A, \frac{B + \sqrt \Delta}{2} \right]. $$
The four ideals are
$$ \left[ 1, \frac{1237 + \sqrt \Delta}{2} \right] , \; \;  \left[ 3, \frac{1237 + \sqrt \Delta}{2} \right] , \; \; \left[ 71, \frac{1221 + \sqrt \Delta}{2} \right] , \; \; \left[ 149, \frac{1221 + \sqrt \Delta}{2} \right] ,    $$
where $\Delta = 1533157.$
