On isogenies of elliptic curves (Q2709930)

From a result of \textit{A. Silverberg} [J. Pure Appl. Algebra 77, 253-262 (1992; Zbl 0808.14037)], one deduces that if two elliptic curves \(E_1\) and \(E_2\) are defined over the number field \(K\), don't have complex multiplication and are isogenous over two different quadratic extensions of \(K\), then \(E_1\) and \(E_2\) are isogenous already over \(K\). The author shows that this is not the case for elliptic curves with complex multiplication by giving an example and showing, afterward, that this example is a typical one. So in this case one can show that if \(E_1\) and \(E_2\) are isogenous over three different quadratic extensions over \(K\), then they are isogenous over \(K\).