Let $a,b\in \Bbb Z, \tau \in \Bbb Q(\sqrt[]{−d})$ where $−d$ is a fundamental discriminant.
Let $N := Norm(aτ + b)$ where $aτ + b$ is an algebraic integer such that $(aτ + b)/m$ is not an algebraic integer for any $m\in \Bbb Z_{> 1}$
Prove that $d\le N$ if $−d \not ≡ 1 (mod 4)$ and $d\le 4N$ otherwise.
I know that $disc(Q(\sqrt[]{−d}))= -d$ if $−d ≡ 1 (mod 4)$ and that $disc(Q(\sqrt[]{−d}))= -4d$ if $−d \not ≡ 1 (mod 4)$ and that there is a bijection between integral quadratic forms and quadratic number fields but I don't see how.
Thank you for your help