Cyclic Quintic Polynomial Family

Question

Is there another known cyclic quintic family other than Lehmer's simplest quintic, which is well known? For instance, in this pdf, in section (4.2) when the parameters $(s_1, s_2, s_3, s_4)$ are set to $s_1=n+1, s_2=n+2, s_3=1, s_4=n+3$, Lehmer's simplest quintic is obtained. What is the two parameter polynomial when $s_1=n-1, s_2=-1, s_3=-1, s_4=n$? Thanks in advance.

Answer 1

Such families are easily construct able now.

In this paper, specializing the parameters $v = 2n+1, w = 1, u = 0$ leads to

$p = n^4 + 2n^3 + 14n^2 + 13n + 11$

Certain linear transformations of the Gaussian Quintic of conductor $p$ lead to the family

$x^5 + (n^2 + n + 4)x^4 + (-2n^2 - 2n + 2)x^3 + (-n^4 - 2n^3 - 10n^2 - 9n - 5)x^2 + (-3n^2 - 3n - 2)x + n^2 + n + 1$

The discriminant of this polynomial is

$(2n + 1)^4(n^2 + n + 1)^2p^4$