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Let $\zeta_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ anarbitrary odd prime. The ratio of Kummer's first factor of the class number ofthe cyclotomic number field $\mathbb{Q}(\zeta_q)$ and its expected order ofmagnitude (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 isbelieved 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 recordmaximum 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, arecollected 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 jointproject with Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova.

Computation of the Kummer ratio of the class number for prime cyclotomic fields