Torsion of elliptic curves with rational 𝑗-invariant defined over number fields of prime degree

Let [ K : Q ] = p [K:\mathbb {Q}]=p be a prime number and let E / K E/K be an elliptic curve with j ( E ) ∈ Q j(E) \in \mathbb {Q} . We determine the all possibilities for E ( K ) t o r s E(K)_{tors} . We obtain these results by studying Galois representations of E E and of its quadratic twists.