  Let $\zeta_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an
arbitrary odd prime. The ratio of Kummer's first factor of the class number of
the cyclotomic number field $\mathbb{Q}(\zeta_q)$ and its expected order of
magnitude (a simple function of $q$) is called the Kummer ratio and denoted by
$r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is
believed that it is unbounded, but only large on a very thin sequence of primes
$q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of
$O(q\log q)$ products and $O(q)$ logarithms. Using it we obtain a new record
maximum for $r(q)$, namely $r(6766811) =1.709379\dotsc$ (the old record being
$r(5231)=1.556562\dotsc$). The program used and the results described here, are
collected at the following address
\url{http://www.math.unipd.it/~languasc/rq-comput.html}.
  This is a (preliminary) report about the computational part of a joint
project with Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova.
