Parametric Solvable Septics?

Question

Known parametric solvable septics are,

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

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

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

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

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

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,

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

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)?

Answer 1

There is a parametric family of cyclic septics that obey $$x_1 x_2 + x_2 x_3 + \dots + x_7 x_1 - (x_1 x_3 + x_3 x_5 + \dots + x_6 x_1) = 0\tag1$$ as the Hashimoto-Hoshi septic, $$\small x^7 - (a^3 + a^2 + 5a + 6)x^6 + 3(3a^3 + 3a^2 + 8a + 4)x^5 + (a^7 + a^6 + 9a^5 - 5a^4 - 15a^3 - 22a^2 - 36a - 8)x^4 - a(a^7 + 5a^6 + 12a^5 + 24a^4 - 6a^3 + 2a^2 - 20a - 16)x^3 + a^2(2a^6 + 7a^5 + 19a^4 + 14a^3 + 2a^2 + 8a - 8)x^2 - a^4(a^4 + 4a^3 + 8a^2 + 4)x + a^7=0$$

For example, let $a=1$ so, $$1 - 17 x + 44 x^2 - 2 x^3 - 75 x^4 + 54 x^5 - 13 x^6 + x^7=0$$ which is the equation involved in $\cos\frac{\pi k}{43}$. If we order its roots as, $$x_1,\,x_2,\,x_3,\,x_4,\,x_5,\,x_6,\,x_7 =\\ r_1,\,r_2,\,r_5,\,r_6,\,r_3,\,r_7,\,r_4 = \\ -0.752399,\; 0.0721331,\; 2.63744,\; 3.62599,\; 0.480671,\; 6.29991,\; 0.636246$$ where the $r_i$ is the root numbering in Mathematica, then it satisfies $(1)$.