Let $K$ be a local field whose residue field's is not characteristic $2$, and $R$ be it's ring of integers, and $M$ be $R$'s maximal ideal. Let $j∈R$, then,
I would like to prove $256(1-λ(1-λ))^3-j λ^2(1-λ)^2＝0$ implies $λ∈R$ and $λ$ is not congruent to $0$ or $1$ mod $M$.
The latter part is obvious, because if $λ$ is congruent to $0$ or $1$, $256＝0$ does not hold in it's residue field.
But I'm stuck with proving $λ∈R$.
Silverman reads we use integrality of $j$, but I don't see how to do it.
Thank you in advance.
This is a question from Silverman's 'the arithmetic of elliptic curves', $p199$.
P.S
$v(j)＝0$ implies $3v(1-λ(1-λ))＝2v(λ)＋2c(1-λ)$, but I cannot proceed from here.