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Mmm sorry I did not understand what you told me, shall I take primes from $\mathbb{Q}(\zeta_5)$ and make them split completely in $E$? And I don't know what is the S-unit group.

By working with eliptic curves, I found that the extension E defined by:

E.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 + 18*x^17 - 32*x^16 + 88*x^15 + 58*x^14 -
                    782*x^13 + 1538*x^12 + 1348*x^11 - 466*x^10 - 894*x^9 + 346*x^8 -
                    114*x^7 - 424*x^6 - 88*x^5 + 214*x^4 + 54*x^3 + 14*x^2 + 4*x + 1)

Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, there exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that $E=\mathbb{Q}(\sqrt[5]{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that the polynomial $X^5-\gamma$ also defines $E$?

Ok, what is happening is that minoply returns de minimal polynomial over $\mathbb{Q}$ and not over a specific field. This answered me. by the way, what

f = G(f)

means?

I want to calculate the relative discriminant of field extensions of this kind: $$\mathbb{Q}(\zeta_5)(\sqrt[5]{a})$$ Where $a \in \mathbb{Q}(\zeta_5)$. So I use SAGE and make this calculations:

K.<b>=CyclotomicField(5);    //my field base
alpha=1+3*b^2;               //an element of my field base
f=(1+3*b^2).minpoly();       //its minimal polynomial
f.is_irreducible()           //is it irreducible?
R.<a>=K.extension(f)         //the field extension of my field base
R.relative_discriminant()    //the calculation of the relative discriminant

But when I execute it, appears this error

defining polynomial (x^4 - x^3 + 6*x^2 + 14*x + 61) must be irreducible

But it is irreducible, what am I doing wrong? Or how can I solve this?