I am looking at the first two examples in Paul Garretts notes on p-adic number theory.
The first example is computing the Newton polygon of $x^5+2x^2+5$ over $\mathbb{Q}_2$. I think this is the lower convex hull of $(0,1),(1,0),(2,0.5),(3,0),(4,0),(5,1)$. So I thought the Newton polygon should just be a horizontal line segment going from $(0,1)$ to $(5,1)$. Of course, I am completely wrong, but I am not sure what is the source of error. Are the points whose convex hull I am supposed to consider correctly identified?