Class Group of the Class Number $3$ with their elements given explicitly INTRODUCTION
The  two-faced behavior of quadratic form $x^2 + 5y^2$  has a hidden companion - the quadratic form $2x^2 + 2xy + 3y^2$ - whose prime values are of the form $20n + 3$ or $20n +7$ (determinant $5$, has two equivalence classes, or class number $2$, irregular behavior).
If we denote the form $x^2 +5y^2$ by $A$ and the form $2x^2 + 2xy + 3y^2$ by $B$, then
Lagrange's results (combined with Brahmagupta's) say that the composites
of $A$ and $B$ have the following "multiplication table":
$$A^2 =A, AB=BA =B, B^2 =A.$$
We recognize this as the multiplication table for the two-element group
with identity element $A$.
Today it is called the class group for $\mathbb Q(\sqrt-5)$.
WHAT I AM LOOKING FOR: A list of quadratic forms with their equivalence classes $A, B, C$, i.e. the class group of the class number $3$ with their elements given explicitly.
I came to know that, there are $25$ such quadratic forms from the paper  "Representation of primes by the principal form of $-D$ when the Class-number $h(- D)$ is $3$". But due to my lacking of technical knowledge in this topic, I can not find the equivalence classes for a specific quadratic form.
probably the following theorem says something about what I am looking -

But I can not decode it, can anyone plz decode it for me, in general if there is  list of quadratic forms with their equivalence classes $A, B, C$,in the above paper, can anyone translate that into an elementary way (like the INTRODUCTION)? Thanks.
EDIT

*

*A deleted answer: The proposition p.132 gives you the list of values of $D$ such that $h(-D)=3$.
So for each $D$, you have to find the corresponding list of reduced forms.

Now $ax^2+bxy+cy^2$ is reduced of discriminant $-D$ if $a,b,c$ are coprime , $|b|\leq a\leq c$ and $b\geq 0\text{ if either }|b|=a\text{ or }a=c$, and of course $b^2-4ac=-D$. The last condition easily implies that $a\leq \sqrt{D/3}$,   so you just have to solve for each value of $D$ by trial and error the finitely many possible values for $a$ abd $b$ (and then $c$)or by programming your favorite CAS.
 A: Here is a Mathematica function that returns all reduced forms of a given discriminant:
reducedForms[d] := 
 Module[{}, 
  Select[Flatten[
            Table[{a, b, (b^2 - d)/(4 a)}, 
                  {a, 1, Floor[Sqrt[-d/3]]}, 
                  {b, Select[Range[-a + 1, a], Mod[#^2 - d, 4 a] == 0 &]}]
         , 1], 
      GCD[Sequence@@#] == 1 && 
      #[[1]] <= #[[3]] && 
      (#[[2]] >= 0 || 4 #[[1]]^2 < #[[2]]^2 - d) &
   ]
  ]

A: There is a copy of part of the list for Class number one here. If discriminant $\Delta = -D$  with positive integer $D,$ when $D \equiv 3 \pmod 8$  we have
$\Delta \equiv 5 \pmod 8$ and $$h(4 \Delta) = = 3 h(\Delta) $$ So
$$h(-44) = 3 h(-11) \; , \; \;$$
$$h(-76) = 3 h(-19) \; , \; \;$$
$$h(-108) = 3 h(-27) \; , \; \;$$
$$h(-172) = 3 h(-43) \; , \; \;$$
$$h(-268) = 3 h(-67) \; , \; \;$$
$$h(-652) = 3 h(-163) \; , \; \;$$
In Buell's Theorem 7.4, when $\Delta < -4,$   we take $s=1.$  Furthermore, page 113, we get $\chi_\Delta(p) = 0$  when $\Delta \equiv 0 \pmod p,$  otherwise
$\chi_\Delta(p) = (\Delta|p)$ is the Jacobi symbol.
Thus, when $\Delta < -4$  and
$\Delta \equiv 0 \pmod p,$
we get
$$h( \Delta p^2) =   h(\Delta) p \; . \; $$
When $\Delta < -4$  and
$\Delta \neq 0 \pmod p,$
we get
$$h( \Delta p^2) = h(\Delta) \left( p - (\Delta|p) \right)  \; . \; $$
In particular,
$$ \color{red}{ h(-243) = 3 h(-27) \; . \; \; }$$


