Let p be a prime number, u relatively prime to p, and $\Lambda := \lbrace (a, b) \in \mathbb{Z}^2 : b \equiv au$ (mod p)$\rbrace$. How then can I find an integral basis $v_1, v_2$ for $\Lambda$?
I need to find this in order to compute the covolume, which is supposed to be the determinant of an integral basis, but it's not clear to me how to get a basis out of this sort of definition for the lattice. I'd like an answer that I could extrapolate out to other lattices defined in terms of equations modulo p, if possible.
(Also, I'm uncertain about the tags here; algebraic number theory is where this problem has come up, but I don't know if it strictly belongs to that field or to some other)
An integral basis is $(1,u)$ and $(0,p)$.
The lattice generated by these vectors contains $(p,0) = p \cdot (1,u) - u \cdot(0,p)$, so clearly contains $p\mathbb{Z} \times p\mathbb{Z}$. If $(a,b)$ is in $\Lambda$ with $p \nmid a$, then let $c$ be an integer with $ac \equiv 1 \pmod{p}$. Then $(a,b) = a \cdot (1,cb) + r \cdot (0,p)$ for some integer $r$. Furthermore, $(1,cb) \in \Lambda$, since $ua \equiv b \pmod{p}$ implies $u \equiv cb \pmod{p}$. Thus, $(a,b)$ is in the sublattice generated by $(1,u)$ and $(0,p)$, which is therefore the whole lattice.