Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define $$ V = \{x \in \mathbb R^6 \mid A \cdot x = 0\} $$ and $$ \Lambda = \{x \in \mathbb Z^6 \mid A \cdot x = 0\}. $$ Equip $\mathbb R^6$ with the standard Euclidean metric and let $V$ inherit that metric from $\mathbb R^6$. Is there a sensible notion of the isomorphism class of the pair $(V, \Lambda)$? And how would this notion depend on the choice of the metric? My apologies if this is elementary or well-known.

Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define $$ V = \{x \in \mathbb R^6 \mid A \cdot x = 0\} $$ and $$ \Lambda = \{x \in \mathbb Z^6 \mid A \cdot x = 0\}. $$ Equip $\mathbb R^6$ with the standard Euclidean metric and let $V$ inherit that metric from $\mathbb R^6$. Is there a sensible notion of the isomorphism class of the pair $(V, \Lambda)$? And how would this notion depend on the choice of the metric?