How to I show that $3x^{2} + xy + 4y^{2}$ and $3x^{2} - xy + 4y^{2}$ are not equivalent over $\mathbb{Z}$. They have the same discriminant so I can't use the contrapositive statement of equivalence $\Rightarrow$ same discriminant.
I was thinking of finding a number which one of them represent and show that the other form can't represent it, but it seems quite hard. Any help is appreciated!
If $A$ and $B$ are equivalent then there exist integers $r,s,t,u$ such that $$B(x,y)=A(r x+ s y,t x + u y)$$ $r,s,t,u$ are the entries of a matrix $M$ in $\text{SL}_2(\mathbb{Z})$. Now, $B(1,0)=A(r,t)=3$ and $B(0,1)=A(s,u)=4$ so $r,t$ and $s,u$ must satisfy the equations $$(6r+t)^2+47t^2=36$$ and $$(6s+u)^2+47u^2=48$$ From the first equation $t=0$ and $r=\pm 1$. From the second equation $u=\pm 1$ and $s=0$. The only possibilities for the matrix $M$ are $I$ or $-I$. Neither one of these transforms $A$ to $B$. Therefore $A$ and $B$ are not equivalent.