Constant root number on integer fibres of elliptic surfaces.

Rizzo showed that the family $\mathcal{W}_t :y^2=x^3+tx^2-(t+3)x+1$, a well-known example of Washington, is such that the root number is $W(\mathcal{W}_t)=-1$ for all $t\in\mathbb{Z}$. In this paper, we fully determinate the non-isotrivial 1-parameter families of elliptic curves $\mathcal{E}_t$ with coefficients (in parameter $t$) of small degree such that the root number is the same for all $\mathcal{E}_t$ with $t\in\mathbb{Z}$.