For my practice implementation of the RSA cryptosystem I use a random value generation function such the value of public key e holds:
The public exponent $e$ and the modulus $n$ should be $e<\phi(n)$
$e$ and $\phi(n)$ are coprime, $gcd(e,\phi(n)) = 1$
Although this works satisfactorily, the algorithm returns values of $e$ which are very large in magnitude which, apparently, is undesirable.
Why is it preferred to generate smaller values for the public exponent in the RSA system?