On the torsion growth in quadratic number fields for elliptic curves defined over the rationals

Abstract Given an elliptic curve defined over the field of rational numbers, it is known how its torsion subgroup may grow when we make a base change to a quadratic number field. In this paper we consider the inverse question: if we have the elliptic curve defined over the rationals and we know how the torsion subgroup grows, what can we say about the field? Our main result gives an explicit relationship between the primes dividing the conductor of the curve and the conductor of the extension as a first approach to a better understanding of this problem.