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From the discriminant of $f$ you know that $\gamma$ is an S-unit where S contains the primes above 5,11. You can compute the relevant S-unit group. Since $\gamma$ is only determined modulo 5th power, this already gives you a finite list of candidates. The list might be a bit long to loop through, so you might want to use some linear algebra and reduction information to cut down the list a little bit.

You can determine a large list of primes that split completely in your field E quite easily (just loop through the primes and keep the ones where the reduction of your given $f$ has a root). At each of those primes your $\gamma$ needs to reduce to a fifth power. That gives you a linear condition mod 5 on the exponent vector wrt. your generating set for S-units. This quite quickly cuts down the list to find that a $\gamma$ with minimal polynomial

x^2-1661*x-161051

has the property you're looking for (that's to say: x^10-1661*x^5-161051 has a root in E).