Torsion growth of rational elliptic curves in sextic number fields

Abstract

We classify the possible torsion structures of rational elliptic curves over sextic number fields. Among these possible torsion group structures, all groups except C3⊕C18 are known to appear as subgroups of E(K)tors for some elliptic curve E/Q and for some sextic number field K. We prove that if the image of mod 2 Galois representation of E is not equal to the Borel subgroup of GL2(Z/2Z), then E(K)tors can't contain the group C3⊕C18.