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Let $f$ and $g$ be two different newforms without complex multiplicationhaving the same coefficient field. The main result of the present articleproves that a congruence between the Galois representations attached to $f$ andto $g$ for a large prime $p$ implies an isomorphism between the endomorphismalgebras of the abelian varieties $A_f$ and $A_g$ attached to $f$ and $g$ bythe Eichler-Shimura construction. This implies important relations betweentheir building blocks. A non-trivial application of our result is that for allprime numbers $d$ congruent to $3$ modulo $8$ satisfying that the class numberof $\mathbb{Q}(\sqrt{-d})$ is prime to $3$, the equation $x^4+dy^2 =z^p$ has nonon-trivial primitive solutions when $p$ is large enough. We prove a similarresult for the equation $x^2+dy^6=z^p$.

Asymptotic results for the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$