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Let $f$ be a quartic polynomial with integer coefficients. For each prime $p$, the ramification type of $f$ modulo $p$ is how it splits modulo $p$. For example, $(4)$ means that $f$ is irreducible modulo $p$, and $(1^2 1^2)$ means that $f \equiv L_1^2 L_2^2 \pmod{p}$ for two linear forms $L_1, L_2$.

It can be shown that if the ramification type of $f$ is $(1^2 1^2), (2^2),$ or $(1^4)$, then the discriminant $\Delta(f)$ is divisible by $p^2$. However, the converse is not true.

My question is, is there an intrinsic way to tell apart those primes $p$ such that $p^2 | \Delta(f)$ and such that the splitting type of $f$ modulo $p$ is a square (that is, $(1^4), (2^2), (1^2 1^2)$)?

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