 Solution

Mod-three Galois representation of an elliptic curve is the representationin this case. Twisting by the quadratic character ${\chi }_d$ replaces it by ${\chi }_d{\rho }_3$. Thus $E^d$ has a rational point of order three exactly when $E[3]$ contains a nonzero vector $v$ satisfyingfor every $\sigma \in G_{\mathbb{Q}}$: the line ${\mathbb{F}}_{3}v$ is a one-dimensional Galois subrepresentation with character ${\chi }_d$.

The two-dimensional representation $E[3]$ has at most two distinct one-dimensional characters among its Jordan–Hölder factors. Therefore at most two quadratic characters ${\chi }_d$, and hence at most two rational isomorphism classes of twists, can have a rational point of order three.