Let $E$ be an elliptic curve, $K$ a number field, $p$ a prime of $\mathbb{Q}$ at which $E$ reduces well.

We know (see for instance Silverman, The Arithmetic of Elliptic Curves, Proposition VIII.1.5(b)) that $K(p^{-1}E(K))/K$ is unramified outside the set consisting of the primes of bad reduction, the primes dividing $p$ and the archmedean places.

My question is towards the reverse: Is there always a place $v$ that is either infinite or of bad reduction and ramifies in $K(p^{-1}E(K))/K$?

It would suffice for my purposes if such a place $v$ existed for all but finitely many $p.$

Let $E$ be an elliptic curve, $K$ a number field so that $\operatorname{rank}_{K}(E)\geq 1,$ $p$ a prime of $\mathbb{Q}$ at which $E$ reduces well.

We know (see for instance Silverman, The Arithmetic of Elliptic Curves, Proposition VIII.1.5(b)) that $K(p^{-1}E(K))/K$ is unramified outside the set consisting of the primes of bad reduction, the primes dividing $p$ and the archmedean places.

My question is towards the reverse: Is there always a place $v$ that is either infinite or of bad reduction and ramifies in $K(p^{-1}E(K))/K$?

It would suffice for my purposes if such a place $v$ existed for all but finitely many $p.$