While perusing Kluener's Database of Number Fields, I noticed that a lot of the discriminants of 7T5 came in pairs. After some doodling, I found four families. The first two are,
$$x^7 - x^6 + x^5 + (n - 1)x^4 - (n + 1) x^3 + 5x^2 - 2x - 1 = 0\tag{1}$$
$$y^7 - y^6 + y^5 - (n + 4)y^4 + (n + 2) y^3 + 5y^2 - 2y - 1 = 0\tag{2}$$
These have the same discriminant,
$$D_1 = (n^2 + 3 n - 29)^2 (27 n^2 + 81 n + 661)^2$$
Eliminating $n$ between (1) and (2), one finds that $x,y$ are related by,
$$-x+x^2-y+x^3y+y^2-x^2y^2+xy^3$$
Another pair is given by,
$$x^7 - x^5 + (n - 1) x^4 - (n + 1)x^3 + 5x^2 - 3x - 1 = 0\tag{3}$$
$$y^7 - y^6 - y^5 + (n + 6)y^4 + (n + 4)y^3 - 7y^2 - 6y - 1 = 0\tag{4}$$
which has discriminant,
$$D_2 = (27 n^4 + 274 n^3 - 967 n^2 - 6454 n - 26017)^2$$
Eliminating $n$ between (3) and (4) will give a different relation between its $x,y$. These are the easy families to find though, from looking at the database, I have a feeling there are more.
Question: Why do a lot of the discriminants/fields of 7T5 come in pairs? Is there anything special about 7T5, or these four families, or is this "pairing" simply an artifact of the search methods/assumptions used by Kluener in creating the database?