Algebraic integers whose conjugates all lie in an ellipse

We find all $15909$ algebraic integers $\boldsymbol {\alpha }$ whose conjugates all lie in an ellipse with two of them nonreal, while the others lie in the real interval $[-1,2]$. This problem has applications to finding certain subgroups of $SL(2,\mathbb{C})$. We use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of $\mathbb{C}$. This gives good bounds for the coefficients of the minimal polynomial of $\boldsymbol{\alpha}.$