While browsing the ***Database of Number Fields***, I came across ***[17T8](http://galoisdb.math.upb.de/search/display?req=p413)***. 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:**


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



What's so special about these $17$th deg equations?
Many of the polynomials in the Klueners-Malle database and also in my
[database with John Jones](http://arxiv.org/abs/1404.0266) 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](http://cda.morris.umn.edu/~roberts/dpr/Research_files/lowgrd.pdf) 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. 


