Let $a, b, c$, pairwise coprime squarefree integers.
Suppose $au^2 + bv^2 + cw^2 ≡ 0 (mod\ 4)$ with $au^2 , bv^2 , cw^2$ pairwise coprime and $(u, v, w) ⊂ \Bbb Z^3$
Let $Λ_0 := \{(x, y, z) ⊂ \Bbb Z^3 : aux + bvy + cwz ≡ 0 (mod\ |abc|)\}$
Let $Λ := \{(x, y, z) ∈ Λ_0 : y ≡ x (mod\ 2) , z ≡ 0 (mod\ 2)\}$.
Prove that if $2 \nmid abc$ and $2|w$ then fundamental domain for $Λ$ has volume $4|abc|$
It is clear that in the fundamental domain we can take: $0<z<2$ and $x<y<x+2$, but how to find an interval for $x$ in order to compute the volume? my intuition is that $0<x<|abc|$ but how to proceed?
Thank you for your help.