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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 $2$.

Question. Is this lattice unique in its genus?

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

Perhaps it is too much asking; to see one example of $k$ such that the lattice $L_k$ is not unique in its genus would also be interesting.

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 $2$.

Question. Is this lattice unique in its genus?

Theorem 21 Chapter 15 of the book "Sphere packing, Lattices and Groups" by Conway and Sloane states that, in order to be not unique, one should have that $4\cdot 18k$ is divisible by $t^3$ for some non-square natural number $t=0$ or $1\, \operatorname{mod}\, 4$. But I would like a general result and to know exactly what is going on for any $k$.

Perhaps it is too much asking; to see one example of $k$ such that the lattice $L_k$ is not unique in its genus would also be interesting.