Poor man's transcendence for Frobenius traces of elliptic curves

Let $E$ be an elliptic curve without complex multiplication defined over $\mathbb{Q}$. Viewing the sequence of its Frobenius traces $(a_p(E){)}_p$ indexed by primes $p$ as an element in the “poor man’s adèle ring”, we prove its transcendence over $\mathbb{Q}$.