  We present a fast algorithm that takes as input an elliptic curve defined
over $\mathbb Q$ and an integer $d$ and returns all the number fields $K$ of
degree $d'$ dividing $d$ such that $E(K)_{tors}$ contains $E(F)_{tors}$ as a
proper subgroup, for all $F \varsubsetneq K$. We ran this algorithm on all
elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves)
and all $d \leq 23$ and collected various interesting data. In particular, we
find a degree 6 sporadic point on $X_1(4,12)$, which is so far the lowest known
degree a sporadic point on $X_1(m,n)$, for $m\geq 2$.
