Suppose I have the Weierstrass Elliptic fibration of $\mathbb{C}$-elliptic curves given as $\mathcal{E}=\{ y^2=x(x-1)(x-\lambda) \} \to \lambda$. The family degenerates to nodal curves at $\lambda = 1$ and $\lambda = 0$.
I'm given automorphisms of the $E_{1/2} = $ fiber over $1/2$, constructed in the following way:
Take 2 loops based at $1/2$, say $\ell_0$ and $\ell_1$ with the first enclosing $0$ and the second enclosing $1$, going counter-clockwise. Following the fibers one gets a "loop" of $E_{\lambda}$ starting and ending at $E_{1/2}$ and so obtain two diffeomorphisms $E_{1/2} \to E_{1/2}$
My question is: how can I see explicitly what these automorphisms are, and how do the two differ, and why did it matter, if it did, on the loops passing through the singular fibers. Of course intuitively this is reminiscent of complex analysis phenomena of defining branch cuts and monodromy business, but I don't know how to formulate it precisely in terms of the family up in $\mathcal{E}$.