Let $E$ be an elliptic curve defined by $E:y^2＝x^3-x$. What is a Galois group $Gal( \Bbb Q(E[4])/ \Bbb Q)$ ? we can easily find $E[2]＝\{(-1, 0), (0, 0), (1, 0), \infty\}$ by finding a point at which tangent line intersects with $E$ at infinity.
But I cannot list up all points of $E[4]$.
Is this difficult to do by hand ? I'm interested in what the Galois group, without listing all points of $E[4]$, are there method to find Galois group ?
Thank you in advance.