Entanglement of elliptic curves upon base extension

Fix distinct primes p and q and let E be an elliptic curve defined over a number field K. The [Formula: see text]-entanglement type of E over K is the isomorphism class of the group [Formula: see text]. The size of this group measures the extent to which the image of the mod [Formula: see text] Galois representation attached to E fails to be a direct product of the mod p and mod q images. In this paper, we study how the [Formula: see text]-entanglement group varies over different base fields. We prove that for each prime [Formula: see text] dividing the greatest common divisor of the size of the mod p and q images, there are infinitely many fields [Formula: see text] such that the entanglement over L is cyclic of order [Formula: see text]. We also classify all possible [Formula: see text]-entanglement types that can occur as the base field L varies.