  Let $E$ be an elliptic curve over $\mathbb{Q}$. Then, we show that the
average analytic rank of $E$ over cyclic extensions of degree $l$ over
$\mathbb{Q}$ with $l$ a prime not equal to $2$, is at most
$2+r_{\mathbb{Q}}(E)$, where $r_{\mathbb{Q}}(E)$ is the analytic rank of the
elliptic curve $E$ over $\mathbb{Q}$. This bound is independent of the degree
$l$ Also, we also obtain some average analytic rank results over $S_d$-fields.
