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 => 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.

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Please clean up your typos in the equations where you multiply: on the left you write u and v but on the right there is a and b instead. –  KCd Mar 30 '11 at 9:25
Every value of C is a value of B since C(x,y) = B(-x,y). –  KCd Mar 30 '11 at 9:49
Notice that the example $x^2+y^2$ works exactly because there is only one reduced form in that class. Maybe you already know this, but I think it could be better to say it explicitly. Regards. –  awllower Mar 29 at 18:54