Given a smooth projective curve $X/\mathbb{Q}_{p}$ of genus $g \ge 2$, there is an induced monodromy action of the Galois group on the $\ell$-adic cohomology of

$$X \otimes \bar{\mathbb{Q_{p}}}.$$

What is an explicit example where the action of wild inertia is non-trivial?

By ``explicit," I meant I would like an example that is not, say, a modular curve. Ideally, $X$ would be, say, presented as an effective divisor on smooth surface. Similarly, I would like to see the monodromy described without using automorphic methods.

(Here $\ell \ne p$ is a prime.)

Advanced topics in the arithmetic of elliptic curvesp. 387) are $y^2=x^3+3$ at $p=3$ and $y^2+2xy=x^3-x^2+2x$ at $2$. To determine the exponent, you can use Ogg's formula or Tate's algorithm (loc. cit.). – François Brunault Mar 28 '12 at 7:56