Generators of the fundamental group of the solid torus

Question

I have a solid torus $T$ and a curve (or knot) $C$ that winds 2-times around the torus (parallel to the longitude). Can I say that $[C] = 2 \in \mathbb{Z} \cong \pi_1(T)$, that is the curve represents 2 in the fundamental group? Can I take the fundamental group without a basepoint, since I want for $C$ do not necessarily go through the basepoint? In a paper I've heard the expression a "conjugacy class of $\pi_1(T)$", if C' represents an element in this conjugacy class, how is it different than $C$?