  Let $k$ be a field and $G$ be a finite group acting on the rational function
field $k(x_g\,|\,g\in G)$ by $k$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in
G$. Noether's problem asks whether the invariant field $k(G)=k(x_g\,|\,g\in
G)^G$ is rational (i.e. purely transcendental) over $k$. In 1974, Lenstra gave
a necessary and sufficient condition to this problem for abelian groups $G$.
However, even for the cyclic group $C_p$ of prime order $p$, it is unknown
whether there exist infinitely many primes $p$ such that $\mathbb{Q}(C_p)$ is
rational over $\mathbb{Q}$. Only known $17$ primes $p$ for which
$\mathbb{Q}(C_p)$ is rational over $\mathbb{Q}$ are $p\leq 43$ and
$p=61,67,71$. We show that for primes $p< 20000$, $\mathbb{Q}(C_p)$ is not
(stably) rational over $\mathbb{Q}$ except for affirmative $17$ primes and
undetermined $46$ primes. Under the GRH, the generalized Riemann hypothesis, we
also confirm that $\mathbb{Q}(C_p)$ is not (stably) rational over $\mathbb{Q}$
for undetermined $28$ primes $p$ out of $46$.
