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LetEλE_{\lambda}be the Legendre family of elliptic curves with equationY2=X⁢(X−1)⁢(X−λ)Y^{2}=X(X-1)(X-\lambda). Given a curve , satisfying a condition on the degrees of some of its coordinates and parametrizing  pointsP1,…,Pm∈EλP_{1},\ldots,P_{m}\in E_{\lambda}and  pointsQ1,…,Qn∈EμQ_{1},\ldots,Q_{n}\in E_{\mu}and assuming that those points are generically linearly independent over the generic endomorphism ring, we prove that there are at most finitely many pointsc0\mathbf{c}_{0}on  such that there exists an isogenyϕ:Eμ⁢(c0)→Eλ⁢(c0)\phi\colon E_{\mu(\mathbf{c}_{0})}\rightarrow E_{\lambda(\mathbf{c}_{0})}and them+nm+npointsP1⁢(c0),…,Pm⁢(c0),ϕ⁢(Q1⁢(c0)),…,ϕ⁢(Qn⁢(c0))∈Eλ⁢(c0)P_{1}(\mathbf{c}_{0}),\ldots,P_{m}(\mathbf{c}_{0}),\phi(Q_{1}(\mathbf{c}_{0})),\ldots,\phi(Q_{n}(\mathbf{c}_{0}))\in E_{\lambda(\mathbf{c}_{0})}are linearly dependent overEnd⁡(Eλ⁢(c0))\operatorname{End}(E_{\lambda(\mathbf{c}_{0})}).

Isogeny relations in products of families of elliptic curves