I have been looking at binary quadratic forms for a question on MSE, [If a binary quadratic form primitively represents $n$
and $n^3$, must it be the identity form?](https://math.stackexchange.com/questions/3985889/if-a-binary-quadratic-form-primitively-represents-n-and-n3-must-it-be-the), about forms representing a prime (not dividing the discriminant) and primitively representing its cube. I calculated that the order of such a form must be one or two or four. The group under Gauss composition, mostly using Dirichlet's description. 

When such a form has nice coefficients as $f=ax^2 + mc xy + ac y^2$, triple $\langle a,mc,ac \rangle,$  some pleasant things happen. The duplicate form has coefficients $\langle a^2,mc,c \rangle$   and the fourth power is evidently the identity, as $c \mid mc$. This is what Dickson calls  "ambiguous" as a  representative of the class.  We define 
\begin{align*}
X  ={} & -ax^3 + 3acx y^2 + mc^2 y^3 \\
Y ={} & mx^3 + 3ax^2 y -ac y^3
\end{align*}
after which
$  F = a X^2 + mcXY + ac Y^2   $ is identically equal to $f^3$ as polynomials in $x$, $y$.

I have a single strange example so far, $\langle 14, 8, 29 \rangle$. It is of order four, and with
\begin{align*}
f ={} & 14 x^2 + 8 xy+29y^2 , \\
u={} &  6x^3 +  60x^2y - 3xy^2  -42y^3,  \\
v ={} & 8x^3- 18x^2y - 60xy^2   +y^3, \\
h ={} & 14 u^2 + 8 uv+29v^2
\end{align*}
cause $h=f^3$ identically as polynomials.
$u$, $v$ are coprime when $x$, $y$ are coprime and
\begin{align*}
y \neq{} & 0 \pmod 2, \\
y \neq{} & x \pmod 3 , \\
x+y \neq{} & 0 \pmod 5 , \\
y-3x \neq{} & 0 \pmod {13}.
\end{align*}

However, I was unable to put the form $\langle 14, 8, 29 \rangle$  into the desired shape $\langle a,mc,ac \rangle$. For a favorable shape, the duplicate $\langle 10, 0, 39 \rangle$
would need to have a representative either $\langle 10, 20v, 39 + 10 v^2 \rangle$ or $\langle 39, 78v, 10 + 39 v^2 \rangle$  where the final coefficient is to be a square, in particular the square of something represented by $\langle 14, 8, 29 \rangle$.  But that does not happen; the proof involves a half dozen Pell type equations. 

> **Question.** Why does $\langle 14, 8, 29 \rangle$ have no equivalent expression as $\langle a,mc,ac \rangle$,  and where might we find
other examples?