Let $A$ be any one of the three elliptic curves over $\mathbb{Q}$ with conductor 11. We show that $A$ has Mordell–Weil rank zero over its field of 5-division points. In each case we also compute the 5-primary part of the Tate–Shafarevich group. Our calculations make use of the Galois equivariance of the Cassels–Tate pairing.