Consider an smooth complex elliptic curve $E$ glued from two affine curves ($p\in\mathbb{C}\setminus0$) $$C_{(x,y)}: y^2=x^3+px\\C_{(s,t)}: t^2=ps^3+s$$ via the coordinate change $s=1/x,t=y/x^2$.
It is well known that principal $\mathbb{G}_a$-bundles up to isomorphism are bijective to elements of $H^1(E,\mathcal{O}_E)$. Using Čech complex we have $$H^1(E,\mathcal{O}_E)=\mathbb{C}\cdot\frac{y}{x}.$$ Choose $\lambda\cdot\frac{y}{x}\in H^1(E,\mathcal{O}_E)$. The corresponding principal $\mathbb{G}_a$-bundle $P_\lambda\to E$ is given by transition function $$g:C_{(x,y)}\setminus\{x=0\}\to \mathbb{G}_a\cong \mathbb{C},\quad (x,y)\mapsto \lambda\cdot\frac{y}{x}. $$
We can exponentiate $g$ to get a holomorphic multiplicative transition function $$\exp(g):C_{(x,y)}\setminus\{x=0\}\to\mathbb{C}^*,\quad (x,y)\mapsto\exp(\lambda\cdot \frac{y}{x}). $$ Then using $\exp(g)$ we get a holomorphic line bundle $L_{\mathrm{hol}}\to E$. Also by GAGA $L_{\mathrm{hol}}\to E$ corresponds to an algebraic line bundle $L_{\mathrm{alg}}\to E$.
Here are my questions.
What is $L_{\mathrm{alg}}\to E$? Can we write down a trivialising cover and algebraic transition functions?
For any principal $\mathbb{G}_a$-bundle on a smooth projective variety, we can exponentiate transition functions to get a $\mathbb{G}_m$-bundle. Do we know the correspondence?