Local factorization of X3 plus 25X2 minus 50X plus 40 ($X^3+25X^2-50X+40$ over ${\mathbb{Q}}_2$ and ${\mathbb{Q}}_5$)

The polynomialsplits into three linear factors over ${\mathbb{Q}}_2$. Over ${\mathbb{Q}}_5$ it is an Eisenstein polynomial, so adjoining a root gives a totally ramified cubic extension. Its splitting field over ${\mathbb{Q}}_5$ has decomposition group $S_3$ and inertia group $A_3\cong C_3$.