Consider the Legendre family $f: X\longrightarrow Y = \mathbb{P}^1\setminus\{0,1,\infty\}$, defined over $K = \mathbb{Q}/\mathbb{Q}_p$.
having fibre above $t\in Y$, the elliptic curve $E_t := y^2 = x(x-1)(x-t)$. Denote by $\mathcal{L} := R^1_{et}f_*\mathbb{Q}_p$ to be the local system associated to the derived pushforward, whose fibre above $t$ is the Etale cohomology $\mathcal{L}_t := H^1_{et}(E_t\otimes\overline{K},\mathbb{Q}_p)$.
This local system has logarithmic singularities at $\{0,1,\infty\}$, and I want to try to understand how the fibres of my local system are defined at tangential basepoints, in particular around $0$ or $\infty$.
For example, for $t = 0$, I imagine these should somehow be related to the etale cohomology of the singular cubic $y^2 = x^2(x-1)$, but it is unclear to me how. Any reference on this topic would be equally appreciated.
