The quadratic forms with discriminant -23 up to change of variables are:
- A(x,y): $x^2 + xy + 6 y^2$
- B(x,y): $2 x^2 - xy + 3 y^2$
- C(x,y): $2 x^2 + xy + 3 y^2$
Viewed as number fields it's relatively easy then compute:
- A(x,y)A(a,b): $A(xa - 6yb,ya + (x - y)b)$.
- BB: $B(xa - (3/2)yb,ya + (x+(1/2)y)b)$
- CC: $C(xa - (3/2)yb,ya + (x-(1/2)y)b)$
I have not found any number of the form C which is not of the form A or B, maybe I just didn't look for enough though.
Can we also compute $A(x,y)B(u,v), AC$ and $BC$?
Is there any way to think about these forms as ideals?
Since 23 = A, 3 = B but 23*3 = B = C so it doesn't seem like there is a simple group structure here, but B and C are 'conjugate' in some sense so perhaps there is a group structure on {{A},{B,C}} or maybe the structure is different than a group?
What about the converse problem? If d|A then d = A,B or C? update 7 is not of the form A, B or C but 7^2 = A.
So in this case the converse problem is not solvable, but I wonder if there are examples of multiple forms where the converse problem does hold?
For an example of a single form where the converse problem works is G(x,y)=$x^2 + y^2$ I have the answer, it's just $d|G \implies d = G$ since every factor of a sum of two squares is a sum of two squares or a square ($= x^2 + 0^2$).
Maybe it would have been better to write this question for discriminant -36 since it has the forms {$x^2 + 9y^2, 2x^2 + 2xy + 5y^2, 3x^2 + 3y^2$} none of which are conjugate... This set seems a bit simpler but it still has the strange non-multiplicative phenomenon with $7^2$.