For $k$ a positive integer, let us consider the rank 3 ternary indefinite lattice $L=L_k$ with quadratic form $$6kx^2-2(y^2+yz+z^2).$$ Its discriminant group has length $2$. My question is: Is that lattice unique in its genus ?

Theorem 21 Chapter 15 of the book "Sphere packing, Lattices and Groups" by Conway and Sloane, tells that in order to be not unique, one should have that $4\cdot 9k$ is divisible by $t^3$ for some nonsquare natural number $t=0$ or $1\, mod\, 4$. But I would like a general result and to know exactly what is going on for any $k$...