$\newcommand{\C}{\mathbb{C}}\newcommand{\Z}{\mathbb{Z}}$ I know from Bott-periodicity that $K_0(C_0(\mathbb{C}))\simeq \Z$, is there any easy way to compute an explicit generator of $K_0(C_0(\mathbb{C}))$? By this I mean a projection $q$ in $M_k(C_0(\mathbb{C})^+)$ such that $[q]-[p_n]$ generates $K_0(C_0(\mathbb{C}))$. Moreover given an element $x\in K_0(C_0(\mathbb{C}))$ is there an easy way to compute $m\in \Z$ such that $m([q]-[p_n])=x$ (this equates to finding the inverse to the isomorphism $\Z\stackrel{\sim}{\rightarrow}K_0(C_0(\mathbb{C}))$)?

I tried digging into the isomorphism $K_0(\C)\stackrel{\sim} {\rightarrow}K_0(C_0(\mathbb{C}))$ that one obtains from Bott-periodicity but things get pretty convoluted. In the end I got a generator that doesn't facilitate any further calculations I'd like to make (like calculating the inverse of the isomorphism).

I'm trying to compute this generator to make some calculations regarding a endomorphism in $K_0(C(\C P^1))\simeq K_0(C_0(\C))\oplus K_0(\C)\simeq \Z\oplus \Z$