TITLE: What's so special about these $17$th deg equations?
QUESTION [15 upvotes]: While browsing the Database of Number Fields, I came across 17T8. It only had four equations, one of which is,
$$\small{x^{17} - 5x^{16} + 40x^{15} - 140x^{14} + 610x^{13} - 1622x^{12} + 4870x^{11} - 10220x^{10} + 22720x^9 - 38080x^8 + 63500x^7 - 84100x^6 + 102200x^5 - 102400x^4 + 83000x^3 \color{brown}{- 55864x^2 + 24080x - 9400}=0}$$
It may not look much, but all four examples had the same coefficients except for the  part in brown (the $x^2, x^1, x^0$ terms), so I assume there might be a parameterization. After some fiddling, I found the rather simple,
$$\frac{(x^5 - 2x^4 + 10x^3 - 10x^2 + 20x - 10)^3\,(x^2 + x + 4)}{(4x^2 - 5x + 25)} = -36m\tag1$$
The four were just the cases $m = -6, -2, -12, 9$. The discriminant of $(1)$ is,
$$D = 2^{36}\, 3^{52}\, 5^{18}\, m^{10}(16m + 81)^8$$
Update: Note that,
$$\frac{(x^5 - 2x^4 + 10x^3 - 10x^2 + 20x - 10)^\color{red}3\,(x^2 + x + 4)}{4x^2 - 5x + 25}-\frac{27^2}{2^2} = \frac{(x-1)(2x^8 - 4x^7 + 32x^6 - 40x^5 + 170x^4 - 136x^3 + 362x^2 - 166x + 185)^\color{red}2}{4x^2 - 5x + 25}\tag2$$
Note that the quintic and octic factors (both irreducible) have solvable Galois groups. Also, I've seen similar factoring behavior in formulas for the j-function like the well-known icosahedral equation,
$$j(\tau)-1728 =-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^\color{red}3}{r^5(r^{10} + 11r^5 - 1)^5}-12^3 = -\frac{(r^{30} + 522r^{25} - 10005r^{20} - 10005r^{10} - 522r^5 + 1)^\color{red}2}{r^5(r^{10} + 11r^5 - 1)^5}\tag3$$
Makes me wonder if $(1)$ is a formula for something. 
Questions:

Does the whole family, except for special $m$, belong to 17T8?
Can one derive it from first principles, instead of a computer search? (Its simple form seem to suggest there might be others.)

REPLY [23 votes]: Many of the polynomials in the Klueners-Malle database and also in my
database with John Jones come from families in the way you correctly
describe.  So you have "reverse engineered" the source family.
This particular source family is the first of two similar families
described in Section 13 my Galois number fields with small root discriminant with Jones.  Your m and our t are related by m=-81/16 t.  The generic
Galois group of this family is 17T8 = SL_2(16).4 and so for almost
all t the specialized Galois group will be 17T8.  Section 13 discusses
some details of the specialization process.  In particular, specializing
at t=-8 keeps the Galois group at 17T8 and has the unusually 
small discriminant 2^16 3^20 5^16.