Smallest pair making a squared plus ab plus b squared square

Among pairs of positive integers $(a,b)$ for which $a^2+ab+b^2$ is a perfect square $c^2$, take the pair with the smallest $a+b$. What is $c^2$?

49. A quick search shows the smallest sum is $a+b=8$, with $\{a,b\}=\{3,5\}$: $9+15+25=49=7^2$. Such pairs can be generated by setting $a^2+ab+b^2=(a-mb)^2$, which gives $b=\tfrac{a(2m+1)}{m^2-1}$; $m=2$ yields $(3,5)$.