Determine ramification at 3 in the non-unique quadratic 3-torsion case

Determine whether the prime 3 ramifies in a quadratic field K over which a rational elliptic curve E acquires a point of order 3 when the field K is not unique.

Background

The paper studies quadratic fields K over which the torsion subgroup of an elliptic curve E defined over \mathbb{Q} grows. For points of order 3, the field producing the torsion growth need not be unique: there can be at most two such quadratic fields. The paper’s results determine the ramification of primes p\geq 5 in these fields, but leave unresolved whether p=3 ramifies.

This unresolved issue arises specifically when E acquires a point of order 3 over more than one quadratic extension. In that situation, the two quadratic fields are related through the 3-division field \mathbb{Q}(E[3]), which contains \mathbb{Q}(\sqrt{-3}); the authors explain that the ramification behavior at 3 cannot be determined by the results developed in the paper.