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Known parametric solvable septics are,


$$x^7 + 21x^5 + 35x^3 + 7x + a(7x^6 + 35x^4 + 21x^2 + 1)=0\tag{2}$$

$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - 1)x^2(x + 1)^2=0\tag{3}$$

$$x^7 + 7x^6 - 7\beta x^2 + 28\beta x + 2\beta(n - 13)=0\tag{4}$$

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

where $\beta = 4(n^2 + 27)$. The first generalizes Demoivre's quintic to 7th powers, the third can be derived from Kluener's database, while the fifth is a variation of the one in this post.

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


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

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

Question: Surely there are other parametric solvable septics, also simple in form, known by now? Can someone give a sixth (without using transformations on the known ones)?

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