Many of the 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.

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.