On the Ramification of Quadratic Fields where there is a torsion growth

Let $E\mathrm{/}\mathbb{Q}$ be a rational elliptic curve whose torsion group grows over a quadratic field $K$. In a previous paper, a relation between the primes of the conductor of $E$ and the primes that divide the discriminant of $K$ was shown. In the present paper, we go further in this study and we compute the ramification of the field $K$, when a torsion point of prime order $\mathrm{\ell }$ is added, in terms of invariants of the curve. Concretely, the conductor, the prime $\mathrm{\ell }$ and the Kodaira symbol at the primes of bad reduction.