Why is the class number $h(-16)=1$?

Question

For an integer $d<0$, we defined the class number of $d$, denoted $h(d)$ as the number of reduced positive definite binary quadratic forms with discriminant $d.$ We quoted a theorem (Baker, Stark 1967) that the only integers $d$ such that $h(d)=1$ are $-3,-4,-7,-8,-11,-19,-43,-67,-163$ - the fundamental discriminants, and $-12,-16,-27,-28$ - not fundamental but still class number 1.

Aren't both $(1,0,4)$ and $(2,0,2)$ reduced +def BQFs with discriminant $-16$? We defined $(a,b,c)$ to be reduced if either $-a<b\leq a<c$ or $0\leq b\leq a=c.$

Answer 1

Example, my little program showing class group and Gauss duplicate of each form. This shows us the principal genus, which is the squares in the group.

There are numerous imprimitive forms for this discriminant.

jagy@phobeusjunior:~/Desktop/Cplusplus$ ./classGroup
Absolute value of discriminant? 
3600
Impr 2 0 450
Impr 3 0 300
Impr 5 0 180
Impr 6 0 150
Impr 10 0 90
Impr 12 0 75
Impr 15 0 60
Impr 18 0 50
Impr 20 0 45
Impr 30 0 30
Impr 4 4 226
Impr 12 12 78
Impr 18 12 52
Impr 24 12 39
Impr 26 12 36
Impr 20 20 50
Impr 25 20 40
Impr 34 32 34
Discr  -3600 = 2^4 * 3^2 * 5^2  class  number  16

 all  
    3600:  < 1, 0, 900>    Square      3600:  < 1, 0, 900>
    3600:  < 4, 0, 225>    Square      3600:  < 1, 0, 900>
    3600:  < 8, -4, 113>    Square      3600:  < 4, 0, 225>
    3600:  < 8, 4, 113>    Square      3600:  < 4, 0, 225>
    3600:  < 9, -6, 101>    Square      3600:  < 9, 0, 100>
    3600:  < 9, 0, 100>    Square      3600:  < 1, 0, 900>
    3600:  < 9, 6, 101>    Square      3600:  < 9, 0, 100>
    3600:  < 13, -12, 72>    Square      3600:  < 25, 0, 36>
    3600:  < 13, 12, 72>    Square      3600:  < 25, 0, 36>
    3600:  < 17, -2, 53>    Square      3600:  < 4, 0, 225>
    3600:  < 17, 2, 53>    Square      3600:  < 4, 0, 225>
    3600:  < 25, -10, 37>    Square      3600:  < 25, 0, 36>
    3600:  < 25, 0, 36>    Square      3600:  < 1, 0, 900>
    3600:  < 25, 10, 37>    Square      3600:  < 25, 0, 36>
    3600:  < 29, -24, 36>    Square      3600:  < 9, 0, 100>
    3600:  < 29, 24, 36>    Square      3600:  < 9, 0, 100>

 squares  
    3600:  < 1, 0, 900>
    3600:  < 4, 0, 225>
    3600:  < 9, 0, 100>
    3600:  < 25, 0, 36>

 fourths  
    3600:  < 1, 0, 900>


Discriminant      -3600     h :   16     Squares :    4     Fourths :    1
jagy@phobeusjunior:~/Desktop/Cplusplus$