Known parametric solvable septics are,

$$x^7+7ax^5+14a^2x^3+7a^3x+b=0\tag{1}$$

$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - 1)x^2(x + 1)^2=0\tag{2}$$

$$x^7 + 7x^6 - 28(n^2 + 27)x^2 + 112(n^2 + 27)x + 8(n^2 + 27)(n - 13)=0\tag{3}$$

$$x^7 + 14x^4 + 7(n - 2)x^3 + 14(n - 5)x^2 - 28x - (n^2 + n + 3)=0\tag{4}$$

The first generalizes *Demoivre's quintic* to 7th powers, the second can be derived from Kluener's database, while the fourth is a variation of the one in this post.

In contrast, *many* parametric solvable quintics are known, such as the multi-variable,

$$x^5+10cx^3+10dx^2+5ex+f=0$$

where the coefficients obey the simple quadratic in $f$,

$$(c^3 + d^2 - c e) ((5 c^2 - e)^2 + 16 c d^2) = (c^2 d + d e - c f)^2$$

* Question*: Surely there are other parametric solvable septics, also simple in form, known by now? Can someone give a

*fifth*(without using transformations on the known ones)?