# About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$:

Up to $\operatorname{SL_2}(\mathbb Z)$ equivalence and scaling, there is only one integral binary quadratic form having an $\operatorname{SL_2}(\mathbb Z)$-automorphism of order three, namely $Q(x,y)=x^2+xy+y^2$.

Now the pertinent definitions in order to understand the problem.