Let $\Lambda$ be a ternary integral lattice in some definite quadratic space $(V,Q)$ over $\mathbb{Q}$.

Consider the usual class set $\text{Cl}(\Lambda) = O(V)\backslash\text{Gen}(\Lambda)$, i.e. the finite set characterising global isometry classes of lattices in the genus of $\Lambda$.

Choose representative lattices $\Lambda_1, \Lambda_2, ..., \Lambda_h$ for the classes in $\text{Cl}(\Lambda)$. Are there (simple?) conditions on $\Lambda$ that guarantee the equality $|\text{Aut}(\Lambda_i)| = |\text{Aut}(\Lambda_j)|$ for all $1\leq i < j\leq h$?

Let $\Lambda$ be an integral lattice in some definite ternary quadratic space $(V,Q)$ over $\mathbb{Q}$.

Consider the usual class set $\text{Cl}(\Lambda) = O(V)\backslash\text{Gen}(\Lambda)$, i.e. the finite set characterising global isometry classes of lattices in the genus of $\Lambda$.

Choose representative lattices $\Lambda_1, \Lambda_2, ..., \Lambda_h$ for the classes in $\text{Cl}(\Lambda)$. Are there (simple?) conditions on $\Lambda$ that guarantee the equality $|\text{Aut}(\Lambda_i)| = |\text{Aut}(\Lambda_j)|$ for all $1\leq i < j\leq h$?