Equivalence of integral primitive indefinite binary quadratic forms

Question

Let $Q_1(x,y)=ax^2+bxy+cy^2$ be an integral primitive indefinite binary quadratic form of positive discriminant $\Delta=b^2-4ac$. From what I deduce from a statement of some text and a calculation it should be $SL_2(\mathbb Z)$-equivalent to $Q_2(x,y)=-cx^2-bxy-ay^2$ in case $c<0$. Can you confirm this? Which matrix gives me the equivalence?

Answer 1

If the discriminant is divisible by a prime $q \equiv 3 \pmod 4,$ then the principal form does not represent $-1,$ and when a form represents some nonzero $a$ it does not represent $-a.$ Which means that $\langle a,b,c \rangle$ and $\langle -c,-b,-a \rangle$ cannot be equivalent. An example is discriminant 21, the principal genus is the class of $\langle 1,3,-3 \rangle,$ with complete cycle

0  form   1 3 -3   delta  -1     ambiguous  
1  form   -3 3 1   delta  3     ambiguous  
2  form   1 3 -3

and the other genus the class of $\langle -1,3,3 \rangle.$

Here is a discriminant with a little more going on:

473    factored   11 *  43

    1.             1          21          -8   cycle length             4
    2.            -1          21           8   cycle length             4
    3.             2          21          -4   cycle length             4
    4.            -2          21           4   cycle length             4
    5.             4          21          -2   cycle length             4
    6.            -4          21           2   cycle length             4

  form class number is   6

one cycle, not principal...

0  form   2 21 -4   delta  -5
1  form   -4 19 7   delta  2
2  form   7 9 -14   delta  -1
3  form   -14 19 2   delta  10
4  form   2 21 -4


  form   2 x^2  + 21 x y  -4 y^2