Let $L'$ be a sublattice of the lattice $[1,\tau]$ in a imaginary quadratic field. Reminder: a lattice $L$ consists of the $\mathbb{Z}$-linear combinations of $1$ and $\tau$, with $\{1,\tau \}$ linear independent over $\mathbb{R}$.
Now, in general a sublattice $L'$ of $L$ is of the form $[a \tau + b, c \tau + d]$.
I need to prove that if
- $[L:L']=|ad-bc|$ is of finite index (=determinant)
- and $q$ is the smallest positive integers contained in $L'$
then $L'$ is of the form $[q, r\tau + s]$ for some suited integers $r$ and $s$.
I did the following... say we have a linear combination
$$ q = \lambda(a \tau + b) + \mu(c \tau +d) = (\lambda a + \mu c) \tau + (\lambda b + \mu d)$$
Then $(\lambda a + \mu c)=0$ implies $\lambda=-kc$ and $\mu=ka$ for some integer $k$.
Now, the equation becomes
$$q = (-kbc+kad) = k \det(ad-bc)$$
Since $q$ must be minimal $k=1$, and $q=|ad-bc|=\det(L')$... not sure how to proceed...